Steering self-organized robot flocks through externally guided individuals

Abstract

In this paper, we study how a self-organized mobile robot flock can be steered toward a desired direction through externally guiding some of its members. Specifically, we propose a behavior by extending a previously developed flocking behavior to steer self-organized flocks in both physical and simulated mobile robots. We quantitatively measure the performance of the proposed behavior under different parameter settings using three metrics, namely, (1) the mutual information metric, adopted from Information Theory, to measure the information shared between the individuals during steering, (2) the accuracy metric from directional statistics to measure the angular deviation of the direction of the flock from the desired direction, and (3) the ratio of the largest aggregate to the whole flock and the ratio of informed individuals remaining with the largest aggregate, as a metric of flock cohesion. We conducted a systematic set of experiments using both physical and simulated robots, analyzed the transient and steady-state characteristics of steered flocking, and evaluate the parameter conditions under which a swarm can be successfully steered. We show that the experimental results are qualitatively in accordance with the ones that were predicted in Couzin et al. model (Nature, 433:513–516, 2005) and relate the quantitative differences to the differences between the models.

Appendix

Mutual Information: Adopting the notation of Feldman [12] and indicating a discrete random variable with the capital letter X, which can take values $x \in {\mathcal{X}}$, the information entropy is defined as:

$$ H[X] = - \sum_{x \in{\mathcal{X}}} p(x) \cdot \log_{2} p(x) $$

where p(x) is the probability that X will take the value of x. H[X] is also called the marginal entropy of X, since it depends on only the marginal probability of one random variable. The marginal entropy of the random variable X is zero if X always assumes the same value with p(X = x′) = 1 and maximum if X assumes all possible states with equal probability. Having defined the marginal entropy of a single random variable, this definition is easily extended to the joint entropy of two random variables:

$$ H[XY] = - \sum_{x \in{\mathcal{X}}} \sum_{y \in {\mathcal{Y}}} p(x,y) \cdot \log_{2}p(x, y) $$

as well as the conditional entropy of these two random variables:

$$ H[X|Y] = - \sum_{x \in{\mathcal{X}}} \sum_{y \in {\mathcal{Y}}} p(x, y) \cdot \log_{2}p(x | y) $$

where p(x, y) is the joint probability that X will take the value of x and Y will take the value of y, and p(x | y) is the conditional probability that X will take the value of x given that Y takes the value of y. Thus, the conditional entropy is the entropy of X, given that Y is known.

Then, the mutual information MI[X, Y] is defined as:

$$ MI[X,Y] = - \sum_{x \in {\mathcal{X}}} \sum_{y \in {\mathcal{Y}}} p(x, y) \cdot \log_{2} {\frac{p(x).p(y)}{p(x, y)}} $$

or equivalently,

$$ \begin{aligned} MI[X,Y] &= H[X] + H[Y] - H[XY]\\ &= H[X] - H[X|Y]\\ &= H[Y] - H[Y|X]. \end{aligned} $$

When p(x|y) becomes 1, the mutual information MI[X, Y] is maximized to be H[X]. Note that, the value of H[X] depends on the discretization of x. For instance, if the value of random variable x is discretized into 8, then p(x) becomes ${\frac{1}{8}}$ leading to $H[X] = -8 \cdot {\frac{1}{8}} \cdot \log_{2} {\frac{1}{8}} = 3.$

Angular Deviation: The angular deviation is calculated as follows [29]. Let θ₁...θ _n denote a set of unit vectors whose angular deviation is to be calculated. Then, their (normalized) mean vector is the vector from (0, 0) to $(\bar{C}, \bar{S})$, where

$$ \bar{C} = {\frac{1}{n}} \sum_{i=1}^{n} \cos \theta_i, \quad \bar{S} = {\frac{1}{n}} \sum_{i=1}^{n} \sin \theta_i. $$

Let $\bar{R} = \sqrt{\bar{S}^2 + \bar{C}^2}$ be the length of this normalized mean vector and $\bar{x}_0$ be its angle with the x-axis such that:

$$ \bar{C} = \bar{R} \cos \bar{x}_0, \quad \bar{S} = \bar{R} \sin \bar{x}_0. $$

Then, the angular deviation of these vectors around their normalized mean vector is given by:

$$ S_0 = 1 - \bar{R}. $$

This intuitively means that the more aligned the vectors are, i.e, the less the angular deviation is, the longer is the mean vector. On the other hand, if they are scattered around the unit circle in a random manner, then their vector sum results in a shorter mean vector, denoting a greater angular deviation from the mean.

The angular deviation around a specific direction can be calculated as an extension of this formulation by letting α denote the angle of the desired direction with the x-axis. Then

$$ \bar{C}^{\prime} = \bar{R} \cos (\bar{x}_0 - \alpha), \qquad \bar{S}^{\prime} = \bar{R} \sin (\bar{x}_0 - \alpha) $$

give the components of the mean vector in the desired direction, and

$$ S_0^{\prime} = 1 - \bar{C}^{\prime} $$

gives the angular deviation around this direction. In the accuracy calculations, we utilize this extended formulation which gives the angular deviation around the desired direction of the flock.