[Paper Review] The rolling problem: overview and challenges

Yacine Chitour, Mauricio Godoy Molina+1 more|arXiv (Cornell University)|Jan 27, 2013

Robotic Mechanisms and Dynamics37 references4 citations

TL;DR

This paper provides a comprehensive overview of the rolling problem—two Riemannian manifolds rolling without slipping or spinning—by tracing its evolution from classical mechanics to modern differential geometry and control theory. The key contribution is the identification of the state space as a principal bundle with a canonical connection, and the proof that the rolling distribution admits a holonomy group structure only when the target manifold has constant curvature, linking geometry and control via curvature invariants.

ABSTRACT

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling model.

Motivation & Objective

• To trace the historical development of the rolling problem from Chaplygin’s early work on non-holonomic systems to modern geometric control theory.
• To establish the geometric structure of the rolling system as a principal bundle with a connection, particularly when the target manifold has constant curvature.
• To investigate the conditions under which the rolling distribution admits a principal bundle structure and how this relates to the curvature of the target manifold.
• To explore the role of symmetries and holonomy in characterizing the controllability and geometric invariants of the rolling system.
• To generalize the rolling model beyond equal-dimensional manifolds and to connect it with Cartan geometries and sub-Riemannian structures.

Proposed method

• Utilizes intrinsic Riemannian geometry and sub-Riemannian geometry to model the rolling dynamics as a distribution on the state space $ Q = Q(M, ilde{M}) $, where $ M $ and $ ilde{M} $ are Riemannian manifolds.
• Applies É. Cartan’s theory of development and affine holonomy to define the rolling connection and analyze its geometric properties.
• Introduces the rolling distribution $ D_{ ext{R}} $ on the state space $ Q $, which encodes non-holonomic constraints of no-slip and no-spin.
• Defines the holonomy group $ ilde{ ext{Hol}}_q(D_{ ext{R}}) $ as the set of transformations preserving the distribution and fiber structure, showing it is isomorphic to the isometry group of $ ilde{M} $ under certain curvature conditions.
• Uses the Lie algebra of symmetries $ ext{Sym}_0(D_{ ext{R}}) $, consisting of vector fields annihilated by the projection $ au: Q o M $, to characterize Killing fields on $ ilde{M} $.
• Establishes a converse result: if a principal bundle structure exists on $ Q $ with $ D_{ ext{R}} $ as a connection, then $ ilde{M} $ must have constant sectional curvature.

Experimental results

Research questions

• RQ1 Under what geometric conditions does the rolling system admit a principal bundle structure with the rolling distribution as a connection?
• RQ2 How is the holonomy group of the rolling distribution related to the isometry group of the target manifold?
• RQ3 What role does the curvature tensor of the target manifold play in determining the controllability and symmetry structure of the rolling system?
• RQ4 Can the rolling model be generalized to manifolds of different dimensions or to Cartan geometries, and what are the implications for controllability?
• RQ5 To what extent do the symmetries of the rolling distribution reflect the intrinsic geometry of the target manifold?

Key findings

• The state space $ Q $ of rolling $ M $ against a space form $ ilde{M} = ilde{oldsymbol{F}}^n_c $ admits a principal $ G_c(n) $-bundle structure with a left action preserving the rolling distribution $ D_{ ext{R}} $.
• The holonomy group $ ilde{ ext{Hol}}_q(D_{ ext{R}}) $ at a point $ q o x o M $ is isomorphic to the isotropy group of the rolling motion and is conjugate to holonomy groups at other points in the same fiber.
• If the curvature operator $ R_x $ is invertible on an open dense subset and the rolling map $ ilde{ ext{Rol}} $ is invertible, then $ ext{Sym}_0(D_{ ext{R}}) $ is isomorphic to the Lie algebra of Killing fields on $ ilde{M} $, implying $ ilde{M} $ has constant curvature.
• The existence of a principal bundle structure on $ Q $ with $ D_{ ext{R}} $ as a connection implies that $ ilde{M} $ must be a space of constant curvature, up to Lie algebra isomorphism.
• The rolling system generalizes naturally to Cartan geometries, unifying rolling of manifolds of different dimensions and pseudo-Riemannian settings under a single framework based on development and connection theory.
• In three dimensions, the orbits of the rolling system can have dimensions 3, 6, 7, 8, or 9, indicating rich geometric and controllability structure.