Distributed matroid-constrained submodular maximization for multi-robot exploration: theory and practice

Abstract

This work addresses the problem of efficient online exploration and mapping using multi-robot teams via a new distributed algorithm for multi-robot exploration, distributed sequential greedy assignment (DSGA), which is based on sequential greedy assignment (SGA). While SGA permits bounds on suboptimality, robots must execute planning steps sequentially. Rather than plan for each robot sequentially as in SGA, DSGA assigns plans to subsets of robots using a fixed number of sequential planning rounds. DSGA retains the same suboptimality bounds as SGA with the addition of a term that describes the additional suboptimality incurred when assigning multiple plans at once. We use this result to extend a single-robot planner based on Monte-Carlo tree search to the multi-robot domain and evaluate the resulting planner in simulated exploration of a confined and cluttered environment. The experimental results show that for teams of 4–32 robots suboptimality due to redundant sensor information introduced in the distributed planning rounds remains small in practice given only two or three distributed planning rounds while providing a 2–8 times speedup over SGA. We also incorporate aerial robots with inter-robot collision constraints and non-trivial dynamics and address subsequent impacts on safety and optimality. Real-time simulation and experimental results for teams of quadrotors demonstrate online planning for multi-robot exploration and indicate that collision constraints have limited impacts on exploration performance.

Appendices

Appendix A: Proof of Theorem 2

Proof

The proof of the suboptimality bound relating DSGA to SGA incorporates suboptimality of the single-robot planner and is similar to (Singh et al. 2009) or (Atanasov et al. 2015). We obtain the following by monotonicity and by rearranging the resulting telescoping sum

$$\begin{aligned} I(M;Y^*)&\le I(M;Y^*) + \sum _{i=1}^{n_r} I\left( M;Y^d_{i}|Y^d_{1:i-1}, Y^*_{i+1:n_r}\right) \nonumber \\&= I(M;Y^d) + \sum _{i=1}^{n_r} I\left( M;Y^*_{i}|Y^d_{1:i-1}, Y^*_{i+1:n_r}\right) . \end{aligned}$$

(25)

By submodularity

$$\begin{aligned} I\left( M;Y^*_{i}|Y^d_{1:i-1}, Y^*_{i+1:n_r}\right) \le I\left( M;Y^*_{i} | Y^d_{1:i-1}\right) . \end{aligned}$$

Without loss of generality, assume that agent indices correspond to the selection order and rewrite in terms of the planning rounds such that $I(M;Y^*_{i} | Y^d_{1:i-1}) = I(M;Y^*_{D_{j,k}}|Y^d_{D_{j,1:k-1}\cup F_{j-1}})$ and note that that although $Y^*$ is formally a set, the mapping from elements to robots can be obtained by the intersections $Y^*\cap \mathcal {Y}_i$ given that the sets $\mathcal {Y}_i$ are disjoint. Then, by submodularity,

$$\begin{aligned} I(M;Y^*_{i} | Y^d_{1:i-1}) \le I\left( M;Y^*_{D_{j,k}}|Y^d_{F_{j-1}}\right) . \end{aligned}$$

By (17) and the greedy maximization step in Algorithm 1

$$\begin{aligned} I\left( M;Y^*_{D_{i,j}}|Y^d_{F_{i-1}}\right) \le \eta I\left( M;Y^d_{D_{i,j}}|Y^d_{F_{i-1}}\right) . \end{aligned}$$

(26)

Substitute (26) and preceding inequalities into (25) to obtain

$$\begin{aligned} I(M;Y^*) \le I(M;Y^d) + \eta \sum _{i=1}^{n_d} \sum _{j=1}^{|D_i|} I\left( M;Y^d_{D_{i,j}}|Y^d_{F_{i-1}}\right) . \end{aligned}$$

(27)

The distributed objective can be rewritten as a sum so that $I(M;Y^d)= \sum _{i=1}^{n_d} \sum _{j=1}^{|D_i|} I(M;Y^d_{D_{i,j}}|Y^d_{D_{i,1:j-1}\cup F_{i-1}})$ and substituted into (27) to obtain

$$\begin{aligned} I(M;Y^*)\le & {} (1+\eta )I(M;Y^d) + \eta \sum _{i=1}^{n_d} \sum _{j=1}^{|D_i|} \left( I\left( M;Y^d_{D_{i,j}}|Y^d_{F_{i-1}}\right) \right. \nonumber \\&\left. -I\left( M;Y^d_{D_{i,j}}|Y^d_{D_{i,1:j-1}\cup F_{i-1}}\right) \right) \end{aligned}$$

(28)

which expresses the suboptimality in terms of decreases in the conditional mutual information from when results are obtained from the planner to when they are assigned. By rewriting mutual information in terms of entropies we can rearrange to obtain the following

$$\begin{aligned} I(M;Y_1) - I(M;Y_1|Y_2) = I(Y_1;Y_2) - I(Y_1;Y_2|M). \end{aligned}$$

If $Y_1$ and $Y_2$ are conditionally independent given M, then the mutual information, $I(Y_1;Y_2|M) = 0$ and so $I(M;Y_1) - I(M;Y_1|Y_2) = I(Y_1;Y_2)$. By substitution into (28) we can obtain the slightly more concise and final expression for the suboptimality in terms of the mutual information between observations

$$\begin{aligned} I(M;Y^*)&\le (1+\eta )I(M;Y^d) \nonumber \\&\quad + \eta \sum _{i=1}^{n_d} \sum _{j=1}^{|D_i|} I(Y^d_{D_{i,j}};Y^d_{D_{i,1:j-1}}|Y^d_{F_{i-1}}). \end{aligned}$$

(29)

$\square$

Appendix B: Proof of Theorem 3

Proof

Equation (21) follows from Theorem 1 by Grimsman et al. (2017) which proves a $k+1$ bound where k is the clique cover number of a directed acyclic graph associated with the planner structure. In this directed graph, each robot represents a vertex, and the graph has a directed edge (a, b) between robots $a,b\in \mathcal {R}$ if b maximizes its objective (15) conditional on the sequence of observations selected by a. Here, a clique, which is a complete subgraph, is analogous to a set of robots that plan sequentially given the choices by all prior robots in the clique. For Algorithm 1, any set of robots $A \subseteq \mathcal {R}$ with at most one robot from each planning round ($|A\cap D_i| \le 1$ for $i \in \{1,\ldots ,n_d\}$) forms a clique in the associated directed graph. A clique cover of size $\lceil n_r/n_d \rceil =\max _{i\in \{1,\ldots ,n_d\}} |D_i|$ can be obtained by selecting cliques with a single robot (as available) from each planning round ($D_1,\ldots ,D_{n_d}$) without replacement. Then, (21) follows by substitution of (17) to obtain a factor of $\eta$. $\square$