Analysis of trade-off between network connectivity robustness versus coverage area of networked multi-robot system

Abstract

This paper analyzes the relationship between connectivity robustification and coverage control of multi-robot systems. A coverage control is a cooperative task of the system to cover a given area by sensors equipped by the robots, and it requires network connectivity to share the sensing information with each other. Regarding network connectivity, network robustification against robot failure is vital since the robots may fail during their team task. Since the network robustification restricts the configuration space of the robots, we have to pay attention to a quantitative trade-off between the coverage area of a networked multi-robot system and the robustness of the network connectivity. Here we report an analysis result of the trade-off using a one-dimensional network model.

Appendix A Derivation of maximum total coverage

We describe the derivation of Eqs. (13, 14, 15). Note that the length of the individual coverage area set $X_{i}$ is 2c in the one-dimensional space, from the definition Eq. (7). Also, note that the distance between two robotic nodes in the coverage subgraph $\mathcal {G}_{C}$ is r and the distance between two robotic nodes in the robustification subgraph $\mathcal {G}_{R}$ is r/2 to maximize the coverage area (see Fig. 3).

We can consider three cases shown in Fig. 8. If $c<r/4$, we obtain

$$\begin{aligned} A(t) = 2c M(t), \end{aligned}$$

(32)

because the intersection of the two coverage area $X_{i}$ and $X_{j}$ is empty for all two nodes $i\ne j$ (see Fig.8 (a)). If $r/4\le c < r/2$, we obtain

$$\begin{aligned} A(t) = 2c V_{C}(t) + \frac{r}{2}(V_{R}+1)+2c, \end{aligned}$$

(33)

because the coverage areas $X_{i}$ in the robustification subgraph $\mathcal {G}_{R}(t)$ are combined together (see Fig.8 (b)). Substituting Eqs. (11, 12) into Eq. (33), we get

$$\begin{aligned} A(t)= & {} \left( 4c-\frac{r}{2} - (4c-r)q \right) M(t) \nonumber \\& + (4c-r)\left( q-s-\frac{1}{2} \right) . \end{aligned}$$

(34)

If $r/2<c$, we obtain

$$\begin{aligned} A(t) = r( V_{C}(t) -2) + \frac{r}{2}(V_{R}+1)+2c, \end{aligned}$$

(35)

because all the coverage areas $X_{i}$ are combined together (see Fig.8 (c)). Substituting Eqs. (11)-(12) into Eq. (35), we get

$$\begin{aligned} A(t) = r \left( \frac{3}{2} - q \right) M(t) + r \left( q-s-\frac{3}{2} \right) + 2c. \end{aligned}$$

(36)

From above all, we get Eqs. (13, 14, 15).

Fig. 8

coverage area of one-dimensional network model with three types of sensor range c