Decentralized dynamic task planning for heterogeneous robotic networks

Abstract

In this paper, we propose a decentralized model and control framework for the assignment and execution of tasks, i.e. the dynamic task planning, for a network of heterogeneous robots. The proposed modeling framework allows the design of missions, defined as sets of tasks, in order to achieve global objectives regardless of the actual characteristics of the robotic network. The concept of skills, defined by the mission designer and considered as constraints for the mission execution, is exploited to distribute tasks across the robotic network. In addition, we develop a decentralized control algorithm, based on the concept of skills for decoupling the mission design from its deployment, which combines task assignment and execution through a consensus-based approach. Finally, conditions upon which the proposed decentralized formulation is equivalent to a centralized one are discussed. Experimental results are provided to validate the effectiveness of the proposed framework in a real-world scenario.

Appendix: A Boolean algebra and matrix operations

A Boolean algebra $\{ \mathbb {B}, \, \otimes , \, \oplus ,\, \lnot , \, 0,\, 1\}$ is a six-tuple consisting of a set $\mathbb {B}$ called universe, equipped with two binary operations $\otimes$ called and, $\oplus$ called or, a unary operation $\lnot$ called complement and two elements $0$ and $1$, such that the following axioms hold: associativity, commutativity, absorption, distributivity and complements.

Let us define a general $n \times m$ logical matrix as $\mathbf {M} \in \mathbb {B}^{n \times m}$, where $\mathbb {B}$ is the boolean set $\{ 0,1\}$ and let us introduce the logical matrix product as follows:

Definition 23

(Logical matrix product) Let us consider two logical matrices $\mathbf {A} \in \mathbb {B}^{n\times m}$, $\mathbf {B} \in \mathbb {B}^{m\times p}$. The logical matrix product $\mathbf {C} \in \mathbb {B}^{n\times p}$ can be defined as: $\mathbf {C} = \mathbf {A} \odot \mathbf {B}$, where each element $\mathbf {C}_{i,j}$ is defined as:

$$\begin{aligned} \mathbf {C}_{i,j} = \bigoplus ^m_{r=1} \mathbf {A}_{i,r} \otimes \mathbf {B}_{r,j} \end{aligned}$$

for each pair $(i,j)$ with $i = \{1,\ldots ,n \}$ and $j = \{1,\ldots , p\}$.

Heres an example to clarify the above definition:

Example

Given the Boolean matrix $\mathbf {A} \in \mathbb {B}^{2 \times 2}$, and the Boolean column vector $\mathbf {b} \in \mathbb {B}^{2}$, defined as follows:

$$\begin{aligned} \mathbf {A} = \left[ \begin{matrix} 1 &{} 0 \\ 0 &{} 1 \\ \end{matrix} \right] , \quad \mathbf {b}= \left[ \begin{matrix} 1 \\ 0 \\ \end{matrix} \right] , \end{aligned}$$

the logical matrix product $\mathbf {c} \in \mathbb {B}^{2}$, is given by

$$\begin{aligned} \mathbf {c}&= \mathbf {A} \odot \mathbf {b} \\&= \left[ \begin{matrix} 1 &{} 0 \\ 0 &{} 1 \\ \end{matrix} \right] \odot \left[ \begin{matrix} 1 \\ 0 \\ \end{matrix} \right] \\&= \left[ \begin{matrix} (1 \otimes 1) \oplus (0 \otimes 0)\\ (0 \otimes 1) \oplus (1 \otimes 0) \\ \end{matrix} \right] = \left[ \begin{matrix} 1 \oplus 0\\ 0 \oplus 0\\ \end{matrix} \right] = \left[ \begin{matrix} 1\\ 0\\ \end{matrix} \right] . \end{aligned}$$