Adaptation is an essential requirement for robotic systems functioning
in dynamic environments that cannot be fully characterized in advance.
Adaptation allows robots to change their behavior in response to
environmental changes and actions of other robots in order to improve
overall system performance. In addition, adaptation allows the system to
remain robust in face of robot failure. Robot memory has been shown to
be a useful mechanism for adaptation. The robots can estimate the global
state of the system from local observations that have been stored in
memory and adjust their behavior accordingly.
We present a mathematical model of the dynamics of collective behavior
of adaptive robot swarms based on the theory of stochastic processes.
Even in a controlled laboratory setting, the actions of an individual
robot are stochastic and unpredictable: the robot is subject to forces
that cannot be known in advance, noise and fluctuations from the
environment, interactions with other robots with complex equally
unpredictable trajectories, errors in its sensors and actuators, in
addition to randomness that is often deliberately inserted into the
robot controller by its designer. We claim that a robot that makes
decisions about future actions based on its memory (and not on any
abstract representations, planning, or higher order reasoning
functions), can be represented by a generalized Markov process. As a
reminder, in a generalized Markov process of order m, a robot’s future
state depends only on m past states. We derive the stochastic Master
Equation, which governs the time evolution of adaptive robot systems.
The Master Equation is often too difficult to formulate and solve for
real systems; therefore, we will work with the Rate Equation, which
represents the first moment of the Master Equation. The Rate Equation
describes how the average number of robots in a given state evolves in time.
We apply our formalism to study adaptive task allocation in mobile
robots. In this application, the robots' task is to forage for Red or
Green pucks (a robot can only forage for one puck type at a time). As
they travel around the arena, robots record observations of pucks and
other robots, and use these observations to estimate the density of
each. If it finds there are not enough robots of a specific type, it may
switch its foraging state to fill the gap. After a transient, we expect
the number of robots in each foraging state to reflect the prevalence of
each puck type in the environment. The Rate Equation describing adaptive
task allocation is a time delay differential equation. We solve the
equation under appropriate initial conditions and show that for some
choice of transition rates, the solutions have a steady state in which
the distribution of robots mirrors the distribution of pucks.
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