By Sadao Kawamura; Mikhail Svinin
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Then, one determines the scalar b from Eq. (12), and substitutes it back into Eq. (11) to obtain the joint velocity. Note, however, that in the vicinity of kinematic singularities this may lead to instability since the determinant of the Jacobian is close to zero. In fact, the above formulation of the inverse kinematics has been developed to deal with motion control around and at kinematic singularities . e. the twist is set to zero. This condition must be Natural Motion and Singularity-Consistent Inversion 13 satisﬁed in order to comply with the existing physical motion constraint at the singularity .
Thus, instabilities related to ill-conditioning can be alleviated. The controller will be referred to as the singularity-consistent kinematic controller. Natural Motion and Singularity-Consistent Inversion 25 DK x − . e . qfb J K + q x (q* ) * disturbance qsc Manipulator + + . qff int b . q . 5 -1 angles [deg] Fig. 7. Singularity-consistent kinematic controller. 5 2 x [m] Fig. 8. Natural motion through a regular point singularity, with joint friction and kinematic closed-loop control. Controller performance will be examined via simulations with the 2R manipulator introduced previously.
Also in 2003, we theoretically proved, without considering the DOF redundancy, the convergence of the grasping motion toward its state of force/torque balance. This research introduced a new concept of the stability on a constraint manifold and showed how it can be used in the veriﬁcation of the grasping stability understood in a dynamic sense. T x = (x, y) y : Endpoint Virtual spring P0 P4 Virtual damper P3 l0 = ||x - xd|| P2 T xd = (xd, yd) : Target P1 x O Torque to generate Virtual spring effect Torque to generate Virtual damper effect Fig.