  A common approach to the provably stable design of reactive behavior,
exemplified by operational space control, is to reduce the problem to the
design of virtual classical mechanical systems (energy shaping). This framework
is widely used, and through it we gain stability, but at the price of
expressivity. This work presents a comprehensive theoretical framework
expanding this approach showing that there is a much larger class of
differential equations generalizing classical mechanical systems (and the
broader class of Lagrangian systems) and greatly expanding their expressivity
while maintaining the same governing stability principles. At the core of our
framework is a class of differential equations we call fabrics which constitute
a behavioral medium across which we can optimize a potential function. These
fabrics shape the system's behavior during optimization but still always
provably converge to a local minimum, making them a building block of stable
behavioral design. We build the theoretical foundations of our framework here
and provide a simple empirical demonstration of a practical class of geometric
fabrics, which additionally exhibit a natural geometric path consistency making
them convenient for flexible and intuitive behavioral design.
