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# Space-Time Continuous Models of Swarm Robotic Systems

Space-Time Continuous Models of Swarm Robotic Systems: Supporting Global-to-Local Programming, by Heiko Hamann

## Fundamentals of Swarm Robotics

Multi-robot systems have the ability to show complex behavior, which is one of the features that motivate my study of them. They also have the potential to solve classic problems in novel, distributed ways. What I’m excited about in this book is the presentation of how we derive partial differential equations (Fokker-Planck equation) from a stochastic differential equation (Langevin equation), which forms the basis of the Brownian motion model.

To get the party started, Hamann lays out a series of definitions we will return to throughout this exploration.

## Brownian motion

Richard Fennman defined Brownian motion as the random motion of particles suspended in a fluid resulting from their collission with the fast-moving molecules in the fluid. What we see are random movements of position for each particle, alternating with displacement to new subregions in the space. If the fluid is in thermal equilibrium, then the fluid’s overal linear and angular momenta remain null of over time. You may be asking yourself what this has to do with robotics, but I think we’ll find that the complex behavior of swarms allows us to conceive of them, at least in one light, as fluid, and thereby we can seek to observe some of the properties already established for fluids (like Bronian motion). If that is indeed possible, what would define thermal equilibrium in a robot swarm?

## Modeling swarms

I found this section to be illustrative, in that Hamann shares how different disciplines have arrived at modeling collective behavior.

### Agent-Based Modeling

This approach is en vogue at the Santa Fe Institute, and has had a long history in computer science, stemming back from von Neumann’s automata, and, later, Langton’s artificial life.

### Control Theory

From the author’s perspective, there’s been work in this space, but the nondeterminism in these models, and how that plays out as coherence breakdowns stemming from local communication, pose a challenge that still needs to be addressed.

### Physics

This is where active matter research is revving up. More on that anon.

### Biology

The big breakthroughs, in my opinion, have come from systems biology. This is where network theory and biology collide. From the author’s perspective, the biggest advances have come from mathematical biology, with the work of Akira Okubo leading the charge. Using robot swarms to replicate biological systems has been explored by the author and Schmickl.