Translating Manipulating Force Polytope by Dynamics on Kinematical Redundant Manipulators

Abstract

In this study, we derive a translation vector of manipulating force polytope. The translation vector is derived a function of velocity and acceleration of extended task space. A dynamic manipulability ellipsoid that is performance index is known to translate by joint velocity [12]. We derived the extended task space constructed by task space and internal motion space, and discovered the dynamic manipulability polytope translate by velocity on the extended task space [13]. However, the effect to the manipulating force polytope by velocity on the extended task space is unclear, because external force by contact doesn't consider. In this paper, we discover that the translation vector of the manipulating force polytope becomes a function of acceleration and tensor production of velocity on the extended task space. The manipulating force polytope spanned on an extended force space constructed by a hand force and a force that cause internal motion. Furthermore, we establish that the manipulating force polytope and the manipulating force ellipsoid indicate to same characteristics by showing the manipulating force polytope is circumscribed to the manipulating force ellipsoid. By the results of this paper, we expect to improve force control performance and to improve collision detection performance on a cooperating robot.

A general method for the manipulability analysis of serial robot manipulators

“…In equation (11), the terms arise due to the end-effector payloads, Coriolis and gravity, and other inertia forces are not considered to ensure the relation has a linear form. When carrying out the dynamic performance analysis, all those ignored terms will be added.…”

“…For a planar robot, its end-effector has a two-dimensional output space, that is, _ p 2 R 2Â1 , F e 2 R 2Â1 , and € p 2 R 2Â1 . From equations ( 9) to (11), it can easily be seen that the feasible regions of the end-effector velocities, forces, and accelerations are bounded by 2n linear inequality constraints coming from the joint velocity and actuated torque limits, respectively. Figure 1 shows the feasible region of a three-DOF planar manipulator.…”

“…Moreover, the final end-effector acceleration polytope is obtained by translating an overall displacement determined by the end-effector payload and the gravity on the basis of the sum polytope constructed by the actuated joint torque constraints and the joint velocity limits as given in equations (11) and (21).…”

“…For a redundantly actuated serial manipulator, let n and m, respectively, denote the number of the actuated joints and the task space dimensions, and n > m. In this case, the manipulator Jacobian J is not a square matrix; thus, the inverse Jacobian J À1 is replaced by the Moore-Penrose pseudo-inverse of the Jacobian J, denoted J y . It is clear that the terms MJ y in equations (11) and ( 16) and M À1 J T in equation ( 16) are n  m matrices in the polytope construction method outlined above. As presented in previous section, the vertices of the end-effector performance polytopes are obtained by solving C m n Á 2 m linear systems of equations and selecting those satisfying all the corresponding constraints.…”

“…Their work focused on the static force capability for redundant manipulators. Okabe 11 investigated the translating manipulability polytope of kinematic redundant manipulator and discovered that the translation vector is a function of velocity and acceleration of the extended task space. Nokleby et al 12 provided a methodology that allows for the actuator limits and includes both analytical and optimization-based methods to determine the force capability of nonredundantly and redundantly actuated parallel manipulators.…”