Geometrical analysis of compliant mechanisms in robotics (euclidean group, elastic systems, generalized springs

摘要

The coordinate free approach to compliant mechanisms in robotics is developed. It is shown that there is no natural positive definite metric on the group SE(3) of rigid body motions and that SE(3) supports a natural family of hyperbolic metrics. Classification of subgroups of SE(3) is used to classify invariant constraints which include lower pairs of mechanism theory as a special case. Stiffness of a generalized spring is intrinsically defined. It is shown that a generic spring stiffness matrix has a normal form which maximally decouples rotational and translational aspects of stiffness. Center of stiffness is defined as the origin of the coordinate frame in which the stiffness matrix assumes this normal form. Analogous results hold for compliance instead of stiffness. Construction of an arbitrary generalized spring by using only stable line springs is described. Elastic systems are defined and their stiffness derived. Equilibrium of an elastic system and its stability are discussed. Holonomic constraints are introduced as limits of very stiff springs. Elastic system with constraints is explicitly described in terms of the Grassmannian formalism. Such systems are shown to be characterized by Lagrangian planes which are naturally reciprocal. Remote center of compliance device, force sensing, and constrained motion are discussed as examples.