Derivation of dual forces in robot manipulators

According to the principle of transference, a compact three-dimensional representation of a rigid body kinematics is obtained by substituting dual for real numbers. This representation has recently been applied to robotics where in addition to its compactness, it allows to constitute the Jacobian matrix explicitly from the product of the dual transformation matrices with no additional computation.This paper introduces the generalized Jacobian matrix. This matrix consists of the complete dual transformation matrices as opposed to the regular Jacobian matrix which consists of specific columns only. The generalized Jacobian matrix relates force and moment at the end-effector to force and moment in all directions, at the joints. It is therefore possible to use the dual transformation matrices to derive, with no additional computation, the full force and moment vector at the robot's joints. Furthermore, the generalized Jacobian matrix also relates motion in all directions at the joints to the motion of the end-effector, an essential relation required at the design stage of robot manipulators (in particular, flexible ones). An extension of these kinematics and statics schemes into dynamics is possible by applying the dual inertia operator as is shown by an example of a three degrees-of-freedom robot.