Generalized Nonlinear and Finsler Geometry for Robotics

Abstract

Robotics research has found numerous important applications of Riemannian geometry. Despite that, the concept remain challenging to many roboticists because the background material is complex and strikingly foreign. Beyond Riemannian geometry, there are many natural generalizations in the mathematical literature-areas such as Finsler geometry and spray geometry-but those generalizations are largely inaccessible, and as a result there remain few applications within robotics. This paper presents a re-derivation of spray and Finsler geometries, critical for the development of our recent work on geometric fabrics [1], which builds the ideas from familiar concepts in advanced calculus and the calculus of variations. We focus on the pragmatic and calculable results, avoiding the use of tensor notation to appeal to a broader audience and emphasizing geometric path consistency over ideas around connections and curvature. It is our hope that they will contribute to an increased understanding generalized nonlinear, and even classical Riemannian, geometry within the robotics community and inspire future research into new applications.

Geometric Fabrics for the Acceleration-based Design of Robotic Motion

“…These experiments represent a global end-effector navigation problem common in many logistics, collaborative, and industrial settings, wherein a robot must autonomously interact with cubbies (bins). We demonstrate the effectiveness of the layer-wise construction of a geometric fabric (other example usage in [1] and [16])) that enables the robot to reach into and out of three sets of cubbies (see Fig. 1).…”

“…The theory of generalized nonlinear geometry and Finsler energy is important for the derivation of geometric fabrics given in [1]. We give just a brief overview of key facts here sufficient for applications of geometric fabrics (see [16] for more details. )…”

“…We require that it be positively homogeneous of degree 2 (HD2), which means that for any λ ≥ 0 we have π(x, λ ẋ) = λ 2 π(x, ẋ). One can show that the HD2 property ensures the differential equation is more than just a collection of trajectories (its integral curves); it additionally has a path consistency property whereby every integral curve starting from a given position x 0 with velocity ẋ0 = η n pointing in a given direction n (here η > 0) will follow the same path [16]. In particular, any variant of the differential equation of the form ẍ = π(x, ẋ) + α(t, x, ẋ) ẋ, where α ∈ R, will have integral curves that trace out the same paths as π.…”

“…Each fabric term defines a triple (M e , f e , π) X , where M e ẍ + f e = 0 derives from the Euler-Lagrange equation applied to L e [16], which can be viewed as two specs, a policy spec [M e , π] X and a natural energy spec (M e , f e ) X . Geometric fabric summation and pullback is, accordingly, defined in terms of the algebra of these two constituent specs.…”

Optimization Fabrics for Behavioral Design

“…In this section, we develop a class of geometric fabric which does not suffer from that problem-energization leaves the system's paths unchanged. See [19] for a detailed overview of the general nonlinear and Finsler geometries used here.…”