Decentralized simultaneous multi-target exploration using a connected network of multiple robots

Abstract

This paper presents a novel decentralized control strategy for a multi-robot system that enables parallel multi-target exploration while ensuring a time-varying connected topology in cluttered 3D environments. Flexible continuous connectivity is guaranteed by building upon a recent connectivity maintenance method, in which limited range, line-of-sight visibility, and collision avoidance are taken into account at the same time. Completeness of the decentralized multi-target exploration algorithm is guaranteed by dynamically assigning the robots with different motion behaviors during the exploration task. One major group is subject to a suitable downscaling of the main traveling force based on the traveling efficiency of the current leader and the direction alignment between traveling and connectivity force. This supports the leader in always reaching its current target and, on a larger time horizon, that the whole team realizes the overall task in finite time. Extensive Monte Carlo simulations with a group of several quadrotor UAVs show the scalability and effectiveness of the proposed method and experiments validate its practicability.

Appendix

For the sake of completeness and readablity, we will recap here the main features of the connectivity maintenance algorithm presented in Robuffo Giordano et al. (2013) with some small changes in the variable names. We start by defining $d_{ij}=\Vert q_i-q_j\Vert$ as the distance between two robot positions $q_i$ and $q_j$, and $d_{ijo}=\min _{\varsigma \in [0,1],o\in \mathcal {O}}\Vert q_i + \varsigma (q_j - q_i) - o\Vert$ as the closest distance from the line of sight between robot i and j to any obstacle.

The main conceptual steps behind the computation of $f_i^\lambda$ can be summarized as follows:

• 1. Define an auxiliary weighted graph $\mathcal {G}^\lambda (t)=(\mathcal {V},\mathcal {E}^\lambda ,W)$, where W is a symmetric nonnegative $n\times n$ matrix whose entries $W_{ij}$ represent the weight of the edge (i, j) and $(i,j)\in \mathcal {E}^\lambda \Leftrightarrow W_{ij}>0$.

• 2. Design every weight $W_{ij}$ as a smooth function of the robot positions $q_i$, $q_j$ and of the obstacle points surrounding $q_i$ and $q_j$, with the property that $W_{ij}=0$ if and only if at least one of the following conditions is verified:

  • (a) the maximum sensing range $R_s$ is reached: $d_{ij} \ge R_s$,

  • (b) the minimum desired distance to obstacles $R_o$ is reached (where $R_o<R_m$): $d_{ijo} \le R_o$;

  • (c) the minimum desired inter-robot distance $R_c$ is reached: $d_{ik} \le R_c$ for at least one $k\ne i$.

• 3. Compute $f_i^\lambda$ as the negative gradient of a potential function $V^\lambda (\lambda _2)$ that grows unbounded when $\lambda _2\rightarrow \lambda _2^\text {min}$ from above, where $\lambda _2$ is the second smallest eigenvalue of the (symmetric and positive semi-definite) Laplacian matrix $L={\text {diag}}_{i=1}^{n}(\sum _{j=1}^{n}W_{ij})-W$, and $\lambda _2^\text {min}$ is a non-negative parameter. This eigenvalue $\lambda _2$ is often also called Fiedler eigenvalue.

It is known from graph theory that a graph is connected if and only if the Fiedler eigenvalue of its Laplacian is positive (Fiedler 1973). If $\mathcal {G}^\lambda (0)$ is connected, and in particular $\lambda _2(0)>\lambda _2^\text {min}$, then under the action of $f_i^\lambda$ the value of $\lambda _2(t)$ can never decrease below $\lambda _2^\text {min}$ and therefore $\mathcal {G}^\lambda (t)$ always stays connected.

From a formal point of view the anti-gradient of $V^\lambda$ for the i-th robot takes the form

$$\begin{aligned} f_i^\lambda =-{\frac{\partial V^\lambda (\lambda _2)}{\partial q_i}}=-{\frac{d V^\lambda }{d \lambda _2}}{\frac{\partial \lambda _2}{\partial q_i}}. \end{aligned}$$

(17)

Moreover, if the formal expression of $V^\lambda$ and W are known then (17) can be analytically computed via the expression (Yang et al. 2010),

$$\begin{aligned} {\frac{\partial \lambda _2}{\partial q_i}}=\sum _{j\in \mathcal {N}_i}{\frac{\partial W_{ij}}{\partial q_i}}(\nu _{2_i}-\nu _{2_j})^2, \end{aligned}$$

(18)

where $\nu _{2_i}$ is the i-th component of the normalized eigenvector of L associated to $\lambda _2$.

In order to have a fully decentralized computation of $f_i^\lambda$, the robots perform a distributed estimation of both $\lambda _2(t)$ and $\nu _{2_i}(t)$, for all $i=1,\ldots ,N$, as shown in Yang et al. (2010). In Robuffo Giordano et al. (2013) the authors finally prove the passivity (and then the stability) of the system w.r.t. the pair $(f_i,v_i)$ for all $i=1,\ldots ,N$, as well as the possibility to compute the connectivity force $f_i^\lambda$ in (17) in a completely decentralized way.