Symbolic Computation of Dynamics on Smooth Manifolds

Abstract

Computing dynamical equations of motion for systems that evolve on complex nonlinear manifolds in a coordinate-free manner is challenging. Current methods of deriving these dynamical models is only through cumbersome hand computations, requiring expert knowledge of the properties of the configuration manifold. Here, we present a symbolic toolbox that captures the dynamic properties of the configuration manifold, and procedurally generates the dynamical equations of motion for a great variety of systems that evolve on manifolds. Many automation techniques exist to compute equations of motion once the configuration manifold is parametrized in terms of local coordinates, however these methods produce equations of motion that are not globally valid and contain singularities. On the other hand, coordinate-free methods that explicitly employ variations on manifolds result in compact, singularityfree, and globally-valid equations of motion. Traditional symbolic tools are incapable of automating these symbolic computations, as they are predominantly based on scalar symbolic variables. Our approach uses Scala, a functional programming language, to capture scalar, vector, and matrix symbolic variables, as well as the associated mathematical rules and identities that define them. We present our algorithm, along with its performance, for computing the symbolic equations of motion for several systems whose dynamics evolve on manifolds such as $\mathbb {R}, \mathbb {R}^{3}, S^{2}, SO(3)$, and their product spaces.