A Simple Control Method Coping With a Kinematically Ill-Posed Inverse Problem of Redundant Robots: Analysis in Case of a Handwriting Robot

Abstract

In order to enhance dexterity in execution of robot tasks, a redundant number of degrees-of-freedom (DOF) is adopted for design of robotic mechanisms like manipulators and multi-fingered hands. Associated with such DOF redundancy relative to the number of physical variables necessary and sufficient for description of a given task, an extra performance index is introduced for controlling such a redundant robot in order to avoid arising of an ill-posed problem of inverse kinematics from the task space to the joint space. This paper treats a handwriting robot as an illustrative example and shows that such an ill-posedness of DOF redundancy can be resolved in a natural way by using a novel concept named "stability on a manifold". It is shown theoretically that sensory feedback signals with a simpler form computed on the basis of measurement data of task-description variables render the closed-loop dynamics to converge asymptotically to a target task-description lying on a lower-dimensional manifold of steady states. Computer simulation concerning specified robot tasks verifies the effectiveness of the proposed control scheme, which results in human-like distribution of joint motions.

Iterative Learning without Reinforcement or Reward for Multijoint Movements: A Revisit of Bernstein's DOF Problem on Dexterity

“…This fact was already discussed in our previous paper [38]. In this paper, we now prove the unique existence of the solution to the pair of 1st-order differential equations of (29) …”

“…In other words, the path of c(t) from q 0 to some point on N P can be called the geodesic on the Riemannian submanifold N(S) induced naturally from the constraint ϕ(x(q)) = 0. Next, consider the full dynamics of the handwriting robot with 4 DOFs shown in Figure 8 by taking into account the effect of gravity forces and damping torques at joints (see [38]). The dynamics is described by…”

A Riemannian-Geometry Approach for Modeling and Control of Dynamics of Object Manipulation under Constraints

“…Fortunately as discussed in the paper [9], it is possible to confirm that a modified scalar function …”

“…Apparently, the Hessian becomes positive definite provided that γ i is chosen as being of similar order of (r i / f d ), and θ i (s i ) remains in a region such that |θ i (s i )| < π/6 for i = 1, 2. Then, by using a similar argument used in proving the stability on a submanifold for DOF redundant systems [9,13], a solution trajectory to the Lagrange equation converges exponentially to the constraint equilibrium manifold. In this paper, we omit the details of the argument, but show a physical meaning of the gradient ∂P/θ together with ∂P/∂p i (i = 1, 2).…”

“…the computed torque method). A naive idea of stability on a manifold by using different metrics for the constrained submanifold and its tangent space was first presented in [9] and used in stabilization control of robotic systems with DOF redundancy [12,13].…”