1. Introduction

Path planning has found practical applications in areas such as entertainment (e.g. robot soccer) [1]; self-driving vehicles (e.g. Google's self-driving cars) [2]; intelligent highways [3], and multiple unmanned space systems [4]. Because of the potential applications, the topic of multipath planning has been studied extensively, for example in [5–11].

The simplicity and potential of consensus algorithms to generate collective behaviors, such as flocking, platooning, rendezvous, and other formation configurations, make it an attractive choice for solving certain problems in multiagent control. However, the basic consensus algorithm collision avoidance mechanism is not developed for adversarial situations (i.e., opposite or attacking motion). To extend the power of the algorithm, it is therefore necessary to develop more powerful collision avoidance capabilities.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Next, we consider the basic approaches to collision avoidance in consensus. Some researchers, for example, [12, 13], approached the avoidance problem by introducing potential forces such as attraction and repulsion. However, the potential force algorithms were not developed for adversarial reconfigurations, for example, vehicles moving in opposite directions. Potential functions also have a problem of getting into local minima, coupled with slow speed of convergence. It is observed in [12] that any repulsion based on potential functions alone is not sufficient to guarantee consensus-based collision avoidance. Moreover, the attitude change maneuver presented in [12] was not developed for three-dimensional space (see [14] for a comprehensive literature survey on this topic).

Thus, in this work, we present an approach which we previously developed [5, 9] for incorporating collision avoidance into the consensus framework by applying quadratically constrained attitude control (Q-CAC), via semidefinite programming (SDP), using linear matrix inequalities (LMI). The main benefit of this approach is that it can solve the collision avoidance problem in adversarial situations and any configurations, and the formulation can be applied to twodimensional as well as three-dimensional spaces. Table 1 shows the notation frequently used in this chapter.


2. Problem statement

Notation Meaning

<sup>v</sup> Distance from <sup>v</sup><sup>i</sup>

z<sup>i</sup> A point on the Z axis of PLi

F Feedback controller matrix K Proportional constant I<sup>p</sup> Identity matrix of size p � p

Γ Γ = L ⊗ I<sup>p</sup>

Table 1. Frequently used notation in this chapter.

N<sup>i</sup> Normal vector perpendicular to x<sup>i</sup>

D Attitude control plant matrix, D ∈S<sup>m</sup> ⊗ Kronecker multiplication operator A State or plant matrix for dynamics of x B Input matrix for dynamics of x for input u

Η A vector or matrix in the Schur inequality R A positive-definite matrix in the Schur inequality Q A symmetric matrix in the Schur inequality

η Positive real number for scaling the consensus term β Positive real number for scaling the proportional term

di

pij

for example in [12, 13]

obstacles with positions x

Consensus is said to have been achieved when <sup>k</sup>xi

Given a set of vehicles i, with initial positions xi

j

The basic consensus problem is that of driving the states of a team of communicating agents to a common value by distributed protocols based on their communication graph. The agents (or vehicles) i(i = 1, ⋯, n) are represented by vertices of the graph, whereas the edges of the graph represent communication links between them. Let xi denote the state of a vehicle i and x is the stacked vector of the states of all vehicles. For systems modeled by first-order dynamics, the following first-order consensus protocol (or its variants) has been proposed,

to l

ij (for 3D) or pij(for 2D)

, vi , and z i

ð Þ<sup>t</sup> <sup>v</sup><sup>i</sup> ð Þ<sup>t</sup> !

http://dx.doi.org/10.5772/intechopen.71288

and xj ð Þ<sup>t</sup> vj ð Þ<sup>t</sup> ! 5

Point of intersection of the lines passing through x<sup>i</sup>

Consensus-Based Multipath Planning with Collision Avoidance Using Linear Matrix Inequalities

� xj

The consensus-based multipath planning with collision avoidance problem can be stated as follows:

Protocol (Eq. (1)) on its own does not solve the collision avoidance problem in adversarial

graph L find a sequence of collision-free trajectories from t<sup>0</sup> to tf such that x<sup>i</sup> tf

k! (xij)

(t0), desired final positions xi

obsð Þ j ¼ 1; ⋯; m , and the Laplacian matrix of their communication

<sup>x</sup>\_ðÞ¼� <sup>t</sup> <sup>L</sup> x tð Þ� xoff : (1)

off, as <sup>t</sup> !∞, <sup>∀</sup><sup>i</sup> 6¼ <sup>j</sup>.

<sup>d</sup>, at time tf, a set of

<sup>¼</sup> xi

<sup>d</sup>∀i.


Table 1. Frequently used notation in this chapter.
