Second-order differential equations define smooth system behavior. In general, there is no guarantee that a system will behave well when forced by a potential function, but in some cases they do and may exhibit smooth optimization properties such as convergence to a local minimum of the potential. Such a property is desirable and inherently linked to asymptotic stability. This paper presents a comprehensive theory of optimization fabrics which are second-order differential equations that encode nominal behaviors on a space and are guaranteed to optimize when forced by a potential function. Optimization fabrics , or fabrics for short, can encode commonalities among optimization problems that reflect the structure of the space itself, enabling smooth optimization processes to intelligently navigate each problem even when the potential function is simple and relatively naive. Importantly, optimization over a fabric is asymptotically stable, so optimization fabrics constitute a  building block for stable system design.
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.
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Second-order differential equations define smooth system behavior. In general, there is no guarantee that a system will behave well when forced by a potential function, but in some cases they do and may exhibit smooth optimization properties such as convergence to a local minimum of the potential. Such a property is desirable and inherently linked to asymptotic stability. This paper
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
R
theory of optimization fabrics which are second-order differential equations that encode nominal behaviors on a space and are guaranteed to optimize when forced by
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
R
Optimization fabrics , or fabrics for short, can encode commonalities among optimization problems that reflect the structure of the space itself, enabling smooth optimization processes to intelligently navigate each problem even when the potential function is simple and relatively naive. Importantly, optimization over a fabric is asymptotically stable, so optimization fabrics constitute
These fabrics shape the system's behavior during optimization but still always provably converge to
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local minimum, making them a
R
for stable system
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