Knife-Edge Motion on a Surface as a Nonholonomic Control Problem

This letter studies a new formulation for the kinematics of a knife-edge moving on an arbitrary smooth surface in <inline-formula> <tex-math notation="LaTeX">${\mathbb {R}^{3}}$ </tex-math></inline-formula>. The kinematics equations for a knife-edge, viewed as a rigid body, are constrained by the requirement that the knife-edge maintain contact with the surface. They describe the constrained translation of the point of contact of the knife-edge on the surface and the constrained attitude of the knife-edge as a rigid body. These equations for the knife-edge kinematics in <inline-formula> <tex-math notation="LaTeX">${\mathbb {R}}^{3}$ </tex-math></inline-formula> are expressed in a geometric form, without the use of local coordinates; they are globally defined without singularities or ambiguities. The kinematics equations can be expressed in several simplified forms and written as a drift-free nonlinear control system. Comments are made about interesting motion planning and path planning problems. The kinematics equations are specialized for two specific surfaces defined in <inline-formula> <tex-math notation="LaTeX">${ {\mathbb {R}}^{3}}$ </tex-math></inline-formula>, namely, a flat plane and the surface of a sphere. Results for the flat plane are compared with standard results obtained using local coordinates; results for the sphere, in contrast, require full attention to the 3-D geometry.