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The principle of transference states that when dual numbers replace real ones all laws of vector algebra, which describe the kinematics of rigid body with one point fixed, are also valid for motor algebra, which describes a free rigid body. No such direct extension exists, however, for dynamics. Rather, the inertia binor is used to obtain the dual momentum, from which the dual equations of motion are derived. This raises the dual dynamic equations to six dimensions, and in fact, does not act on the dual vector as a whole, but on its real and dual parts as two distinct real vectors. Moreover, in order to obtain the dual force as a derivative of the dual momentum in a correct order, real and dual parts have to be artificially interchanged. In this investigation the dual inertia operator, which allows direct relation of dual momentum to dual velocity, is introduced. It gives the mass a dual property which has the inverse sense of Clifford’s dual unit, namely, it reduces a motor to a rotor proportional to the vector part of the former. In a way analogous to the principle of transference, the same equation of momentum and its time derivative, which holds for a linear motion, holds for both linear and angular motion of a rigid body if dual force, dual velocity, and dual inertia replace their real counterparts. It is shown that by systematic application of the dual inertia for derivation of the dual momentum and the dual energy, both Newton-Euler and Lagrange formulations of equations of motion are obtained in a complete three-dimensional dual form. As an example, these formulations are used to derive the inverse dual dynamic equations of a robot manipulator.

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This paper describes a new structure of a six-DOF parallel robot. First, a known planar three-DOF double-triangular structure is modified by replacing the stationary triangle with a circle. It increases the work envelope considerably especially when rotational motions are required. The ability for unlimited rotational motion allows extending the structure into six-DOF by using two sets of stationary circles and moveable triangles. Each set can actuate the moving triangle in a planar three-DOF motion and hence actuate a line connecting the centers of the movable triangles in four-DOF. The robot's end-effector is attached to a link along this line while rotation about and translation along this line are obtained by the additional rotational DOF of the movable triangles. The solution of the direct kinematics of this six-DOF manipulator is given in a closedform and it is shown that at most, four different solutions exist.

This research investigates the stability of planar grasps with a multi-fingered robotic hand, using energy approach and geometric interpretation. A more general non-linear finger model was adopted, which reveals that the conditions for stability, obtained by traditional linearized model, are too relaxed. Geometrically, the critical conditions for the linearized planar model constitute a hyperplane in the space of grasping forces, whereas the non-linearized model constitutes a thirdorder surface contained within the permissible region of the former. Hence, allowable grasping forces calculated by a linearized model may practically lead to instability. The linearized finger model analysis shows that at critical force there is one and only one instantaneous instability center fixed in the plane (which coincides with the compliance center during loading), infinitesimal rotation about which causes instability. When a nonlinearized finger model is considered, the compliance center position depends on the applied forces and it moves in the plane during loading. Furthermore, in some cases, there appear a set of instantaneous instability centers as the critical level of forces is reached.