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Action-Consistent Decentralized Belief Space Planning with Inconsistent Beliefs and Limited Data Sharing: Framework and Simplification Algorithms with Formal Guarantees

Tanmoy Kundu, Moshe Rafaeli, Anton Gulyaev, Vadim Indelman

TL;DR

This work addresses multi-robot belief-space planning when robots hold inconsistent beliefs due to limited data sharing. It introduces action-consistency as a robust criterion and develops VerifyAC, which guarantees MR-AC by accounting for missing observations; when needed, EnforceAC triggers self-initiated communications to reach consistency. To reduce communication and computation, the paper extends VerifyAC with R-VerifyAC and R-VerifyAC-simp, providing deterministic or probabilistic guarantees. The framework is demonstrated in discrete search-and-rescue and continuous active visual-SLAM experiments, showing substantial reductions in communications while maintaining desired coordination, with extensions to high-dimensional spaces via probabilistic estimators. Collectively, the methods enable safe, coordinated decision making under partial observability with constrained data sharing, across both simulated and real-robot scenarios.

Abstract

In multi-robot systems, ensuring safe and reliable decision making under uncertain conditions demands robust multi-robot belief space planning (MR-BSP) algorithms. While planning with multiple robots, each robot maintains a belief over the state of the environment and reasons how the belief would evolve in the future for different possible actions. However, existing MR-BSP works have a common assumption that the beliefs of different robots are same at planning time. Such an assumption is often unrealistic as it requires prohibitively extensive and frequent data sharing capabilities. In practice, robots may have limited communication capabilities, and consequently beliefs of the robots can be different. Crucially, when the robots have inconsistent beliefs, the existing approaches could result in lack of coordination between the robots and may lead to unsafe decisions. In this paper, we present decentralized MR-BSP algorithms, with performance guarantees, for tackling this crucial gap. Our algorithms leverage the notion of action preferences. The base algorithm VerifyAC guarantees a consistent joint action selection by the cooperative robots via a three-step verification. When the verification succeeds, VerifyAC finds a consistent joint action without triggering a communication; otherwise it triggers a communication. We design an extended algorithm R-VerifyAC for further reducing the number of communications, by relaxing the criteria of action consistency. Another extension R-VerifyAC-simp builds on verifying a partial set of observations and improves the computation time significantly. The theoretical performance guarantees are corroborated with simulation results in discrete setting. Furthermore, we formulate our approaches for continuous and high-dimensional state and observation spaces, and provide experimental results for active multi-robot visual SLAM with real robots.

Action-Consistent Decentralized Belief Space Planning with Inconsistent Beliefs and Limited Data Sharing: Framework and Simplification Algorithms with Formal Guarantees

TL;DR

This work addresses multi-robot belief-space planning when robots hold inconsistent beliefs due to limited data sharing. It introduces action-consistency as a robust criterion and develops VerifyAC, which guarantees MR-AC by accounting for missing observations; when needed, EnforceAC triggers self-initiated communications to reach consistency. To reduce communication and computation, the paper extends VerifyAC with R-VerifyAC and R-VerifyAC-simp, providing deterministic or probabilistic guarantees. The framework is demonstrated in discrete search-and-rescue and continuous active visual-SLAM experiments, showing substantial reductions in communications while maintaining desired coordination, with extensions to high-dimensional spaces via probabilistic estimators. Collectively, the methods enable safe, coordinated decision making under partial observability with constrained data sharing, across both simulated and real-robot scenarios.

Abstract

In multi-robot systems, ensuring safe and reliable decision making under uncertain conditions demands robust multi-robot belief space planning (MR-BSP) algorithms. While planning with multiple robots, each robot maintains a belief over the state of the environment and reasons how the belief would evolve in the future for different possible actions. However, existing MR-BSP works have a common assumption that the beliefs of different robots are same at planning time. Such an assumption is often unrealistic as it requires prohibitively extensive and frequent data sharing capabilities. In practice, robots may have limited communication capabilities, and consequently beliefs of the robots can be different. Crucially, when the robots have inconsistent beliefs, the existing approaches could result in lack of coordination between the robots and may lead to unsafe decisions. In this paper, we present decentralized MR-BSP algorithms, with performance guarantees, for tackling this crucial gap. Our algorithms leverage the notion of action preferences. The base algorithm VerifyAC guarantees a consistent joint action selection by the cooperative robots via a three-step verification. When the verification succeeds, VerifyAC finds a consistent joint action without triggering a communication; otherwise it triggers a communication. We design an extended algorithm R-VerifyAC for further reducing the number of communications, by relaxing the criteria of action consistency. Another extension R-VerifyAC-simp builds on verifying a partial set of observations and improves the computation time significantly. The theoretical performance guarantees are corroborated with simulation results in discrete setting. Furthermore, we formulate our approaches for continuous and high-dimensional state and observation spaces, and provide experimental results for active multi-robot visual SLAM with real robots.
Paper Structure (23 sections, 11 theorems, 38 equations, 16 figures, 3 tables, 1 algorithm)

This paper contains 23 sections, 11 theorems, 38 equations, 16 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries & Problem Formulation
  4. Preliminaries
  5. Problem Formulation
  6. Approach
  7. Action preferences with different beliefs
  8. MR-AC for robots with inconsistent beliefs
  9. Algorithm VerifyAC
  10. Self-triggered decision for communication
  11. Relaxation and Simplification of VerifyAC with performance guarantees
  12. Algorithm R-VerifyAC
  13. R-VerifyAC-simp: Simplification of R-VerifyAC
  14. Continuous and High-Dimensional Cases
  15. Observation Sequence Likelihood Estimation
  16. ...and 8 more sections

Key Result

Theorem 4.3

Steps 1-3 of VerifyAC are necessary and sufficient for any robot $r$ to find MR-AC, if MR-AC exists, with the robots in $\Gamma = \{r,r'\}$ having inconsistent beliefs.

Figures (16)

  • Figure 1: (a) Two robots, $r$ and $r'$, acquire separate observations ($z^r, z^{r'}$) and begin a planning session. The robots aim to cooperatively reach the green star while satisfying a safety property of avoiding obstacles and collisions; each agent has two candidate actions ({$a^r, \bar{a}^r$} for robot $r$, and $\{a^{r'}, \bar{a}^{r'}\}$ for robot $r'$). (b) The robot communicate their observations to each other (red color), after which their histories, and beliefs, become consistent. Decentralized MR-BSP in such case yields the same best joint action for both robots. Here, $H^c$ represents the common history between the robots prior to acquiring the observations $z^r$ and $z^{r'}$. (c) The robots do not communicate their observations, and as a result, the robots' beliefs become inconsistent. This causes the robots to conclude inconsistent joint-actions, which leads to a collision.
  • Figure 2: Illustration of $H^r, H^{r'}, \leftidx{^c}{\mathcal{H}}, \Delta \mathcal{H}^{r,r'}$, and $\Delta \mathcal{H}^{r',r}$. See text for details.
  • Figure 3: Illustration of VerifyAC from the perspective of robot $r$. Robots $r$ and $r'$ have inconsistent beliefs $b^r_k$ and $b^{r'}_k$ at time $k$. Candidate joint actions are $\bar{a}$ and $\bar{a}'$. Triangles and squares denote objective function ($J(.)$) evaluations for $r$ and $r'$ respectively. (a) Step 1 of $r$: Robot $r$ computes its belief for its actual observation. Chooses $\bar{a}$ as the best action. (b) Step 1-2 of $r$: In Step 2, $r$ computes $J(.)$ for each possible observation of $r'$. All the observations are consistent in favor of $\bar{a}$. (c) Step 1-2 of $r'$: Similarly, robot $r'$ computes Step 1 for its actual observation, and Step 2 for all possible observations of $r$. (d) Step 3 of $r$: Combines (a)-(c) and verifies that the observations at each step are consistent in favor of action $\bar{a}$. Hence, $r$ can be assured that $r'$ also has chosen $\bar{a}$. Thus $r$ chooses action $\bar{a}$ at time $k$.
  • Figure 4: Flowchart of VerifyAC and the conditions for communications.
  • Figure 5: Illustration of R-VerifyAC from the perspective of robot $r$. Consider, robot $r$ selects action $\bar{a}$ in step 1. Even though all the observations are not in favor of $\bar{a}$, R-VerifyAC provides (a) probabilistic guarantee on MR-AC, or (b) a comm is triggered from $r$ to $r'$. Triangles and squares represent the $J$-values corresponding to the observations in steps 2 and 3 respectively. Let, $1-\epsilon = 0.1$. (a) For action $\bar{a}$, in step 2, $\leftidx{^2}{\mathtt{Cl}_{\bar{a}}} = \leftidx{^2}{\mathtt{Cl}^*} = 0.3+0.5 = 0.8 > 1-\epsilon$; in step 3, $\leftidx{^3}{\mathtt{Cl}_{\bar{a}}} = 0.4+0.3 = 0.7 > 1-\epsilon$. So, according to Definition \ref{['def:epsilon-mrac']}, $\epsilon\textsc{-MRAC}^r = \mathtt{true}$, and according to Theorem \ref{['thm:probabilistic-guarantees-rverifyac']}, robot $r$ will not trigger a comm. Probability of MR-AC with both robots choosing $\bar{a}$ is $\leftidx{^2}{\mathtt{Cl}_{\bar{a}}} = 0.8$; probability of robot $r$ choosing $\bar{a}$ but $r'$ choosing an inconsistent action $a$ ($\neq \bar{a}$) is $0.2$; and probability of $r'$ triggering a comm is $1-0.8-0.2 = 0$. (b) For action $\bar{a}$, in step 2, $\leftidx{^2}{\mathtt{Cl}_{\bar{a}}} = 0.2+0.2 = 0.4 < 1-\epsilon$ and $\leftidx{^2}{\mathtt{Cl}_{\bar{a}}} \not = \leftidx{^2}{\mathtt{Cl}^*}$. So, according to Definition \ref{['def:epsilon-mrac']}, $\epsilon\textsc{-MRAC}^r = \mathtt{false}$, and according to Theorem \ref{['thm:probabilistic-guarantees-rverifyac']}, robot $r$ will trigger a comm from itself to $r'$.
  • ...and 11 more figures

Theorems & Definitions (16)

  • Definition 4.1: Multi-Robot Action Consistency (MR-AC)
  • Definition 4.2: Consistent observations
  • Theorem 4.3
  • Theorem 4.4
  • Definition 5.1: Cumulative likelihood of obs. favoring $a$
  • Definition 5.2: $\epsilon\textsc{-MRAC}^r$
  • Theorem 5.3: $\epsilon\textsc{-MRAC}$ Symmetry
  • Theorem 5.4: Probabilistic guarantees on R-VerifyAC
  • Corollary 5.5: Deterministic guarantee on MR-AC
  • Corollary 5.6: Zero Probability for Action Inconsistency
  • ...and 6 more