Abstract
A Riemannian-geometry approach for modeling and control of dynamics of object manipulation under holonomic or non-holonomic constraints is presented. First, position/force hybrid control of an endeffector of a multijoint redundant (or nonredundant) robot under a holonomic constraint is reinterpreted in terms of “submersion” in Riemannian geometry. A force control signal constructed in the image space of the constraint gradient is regarded as a lifting (or pressing) in the direction orthogonal to the kernel space. By means of the Riemannian distance on the constraint submanifold, stability of position control under holonomic constraints is discussed. Second, modeling and control of two-dimensional object grasping by a pair of multijoint robot fingers are challenged, when the object is of arbitrary shape. It is shown that rolling contact constraints induce the Euler equation of motion, in which constraint forces appear as wrench vectors affecting the object. The Riemannian metric is introduced on a constraint submanifold characterized with arclength parameters. An explicit form of the quotient dynamics is expressed in the kernel space with accompaniment of a pair of first-order differential equations concerning the arclength parameters. An extension of Dirichlet-Lagrange's stability theorem to redundant systems under constraints is suggested by introducing a Morse-Lyapunov function.
Highlights
Among roboticsists, it is implicitly known that robot motions can be interpreted in terms of orbits on a high-dimensional torus or trajectories in an n-dimensional configuration space
This paper first emphasizes a mathematical observation that, given a robot as a multibody mechanism with n degrees of freedom whose endpoint is free, the set of all its postures can be regarded as a Riemannian manifold (M, g) associated with the Riemannian metric g that constitutes the robot inertia matrix
It should be emphasized that once the Riemannian manifold is given corresponding to the n degrees of freedom robot, the collection of all the geodesic paths describes the “law of inertia” for the manifold
Summary
It is implicitly known that robot motions can be interpreted in terms of orbits on a high-dimensional torus or trajectories in an n-dimensional configuration space. An explicit form of the Euler equation whose solution corresponds to a geodesic on the submanifold is given as a quotient dynamics corresponding to the kernel space as an orthogonal compliment to the image space spanned from all the constraint gradients. Based upon these observations, the well-known methodology of hybrid (position/force) control for a robot whose end effector is constrained on a surface is re-examined and shown to be effective even if the robot is of redundancy in its degrees of freedom. A sketch of the convergence proof is given on the basis of an extension of the Dirichlet-Lagrange theorem to a system of degrees of freedom redundancy by finding a Morse-Lyapunov function and using its physical properties and mathematical meanings
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