2. Problem statement

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

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

comprehensive literature survey on this topic).

4 Advanced Path Planning for Mobile Entities

Notation Meaning

x<sup>i</sup> Position vector of vehicle number i

off Offset vector of vehicles i and j

, x\_<sup>i</sup> Control input of vehicle i

L Laplacian matrix

ε Width of safety region

v<sup>i</sup> Attitude vector of vehicle i

obs Obstacle vector of vehicle i

ij Line of intersection of PLi

<sup>x</sup> Distance from <sup>x</sup><sup>i</sup>

<sup>∗</sup> Radius of S r r<sup>∗</sup> + ε

x Stacked vector of more than one position vector xoff Stacked vector of more than one offset vector

<sup>S</sup><sup>m</sup> The set of <sup>m</sup> <sup>m</sup> positive-definite matrices S Bounding sphere or circle of a vehicle or obstacle

u, x\_ Stacked vector of control inputs of more than one vehicle

obs Obstacle vector of vehicle <sup>i</sup> emanating from vehicle <sup>j</sup>

Lij Line passing through the mid points of vehicles i and j ρij Perpendicular bisector of Lij separating vehicles i and j PLi Plane passing through the midpoint of vehicle i

and PLj

(for 2D)

ij (for 3D) or pij

to l

Dij Euclidean distance between vehicles i and j

this chapter.

(xij )

ui

r

vi

v ij

l

di

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, for example in [12, 13]

$$\dot{\mathbf{x}}(t) = -\mathbf{L}\left(\mathbf{x}(t) - \mathbf{x}^{\text{off}}\right). \tag{1}$$

Consensus is said to have been achieved when <sup>k</sup>xi � xj k! (xij) off, as <sup>t</sup> !∞, <sup>∀</sup><sup>i</sup> 6¼ <sup>j</sup>.

The consensus-based multipath planning with collision avoidance problem can be stated as follows: Given a set of vehicles i, with initial positions xi (t0), desired final positions xi <sup>d</sup>, at time tf, a set of obstacles with positions x j obsð Þ j ¼ 1; ⋯; m , and the Laplacian matrix of their communication graph L find a sequence of collision-free trajectories from t<sup>0</sup> to tf such that x<sup>i</sup> tf <sup>¼</sup> xi <sup>d</sup>∀i. Protocol (Eq. (1)) on its own does not solve the collision avoidance problem in adversarial situations. A comprehensive presentation of the necessary mathematical tools for this work (including graph theory and consensus theory) can be found in [14].
