Decentralized stochastic control of robotic swarm density: Theory, simulation, and experiment

Quantifying Robotic Swarm Coverage

“…We apply this idea but with the opposite aim: namely, we represent discrete points (the robots' positions) with a continuous function. A similar strategy was alluded to in [18] and used in [25,40] to measure the effectiveness of a certain robotic control law, but to our knowledge, our work here and in [1] is the first to develop any such method in a form sufficiently general for common use. This section is devoted to our definition of the error metric and to its basic properties and computational considerations.…”

“…Under the stochastic control law of [25], the behavior of the error metric over time appears to be a noisy decaying exponential. Therefore, we fit to the data shown in Figure 7 of [25] a function of the form f (t) = α + β exp(− t τ ) by finding error asymptote α, error range β, and time constant τ that minimize the sum of squares of residuals between f (t) and the data. By convention, the steady state settling time is taken to be ts = 4τ , which can be interpreted as the time at which the error has settled to within 2% of its asymptotic value [3].…”

“…Remark 1. Although the sentiment of the e rel benchmark is found in [25], we have formalized the calculation and made three important improve-ments to make it suitable for general use. First, we have placed the calculation of e − and e + into a framework appropriate for nonlinear programming, so that the calculation is objective and repeatable.…”

“…Therefore, one needs a better understanding of the finer properties of the error metric in order to judge whether its values (and hence the performance of the underlying controller) are "good" or not. We address this by studying two benchmarks, 1. the extrema of the error metric, and 2. the probability density function (PDF) of the error metric when robot positions are sampled from the target distribution, which were first proposed in [25]. Using tools from nonlinear programming for (1) and rigorous probability results for (2), we put each of these benchmarks on a firm foundation.…”

Unstable modes and bistability in delay-coupled swarms

“…Deriving inspiration from nature, embodied artificial swarm systems have been created to mimic emergent pattern formation-with the ultimate goal of designing robotic swarms that can perform complex tasks autonomously [15][16][17][18]. Recently robotic swarms have been used experimentally for applications such as mapping [19], leader-following [20,21], and density control [22]. To achieve swarming behavior, often, robots are controlled based on models, where swarm properties can be predicted exactly [23][24][25][26][27].…”