Dual numbers and dual vectors are widely used in spatial kinematics [3,5-15,18]. Plücker line coordinates of a straight line can be represented by a dual unit vector located at the dual unit sphere (DUS). By this way, the trajectory of the screw axis of a rigid body in 3 R (the real three space) corresponds to a dual curve on the DUS. This correspondence is done through Study Mapping [8,9]. Conversely a dual curve on DUS obtained from the rotations of the DUS represents a rigid body motion in 3 R [8]. The dual Euler parameters are used in defining the screw transformation in 3 R [8], but originally in this paper these parameters are constructed from the Rodrigues and the dual Rodrigues parameters [15].
Roth and Bottema obtain the instantaneous properties of a plane motion and its inverse motion by the instantaneous invariants depending on a canonical frame which was introduced in Veldkamp's dissertation(1963). We obtain the instantaneous invariants depending on the same frame by using Lie algebra of the plane motion.
Under liquid swimming for the robots is extremely interesting. In this context one can imagine deep sea beds, oil deposits, acid tanks, etc. It is believed that the next generation of robots will be based on animals rather than humans. If we consider the underwater swimming robots, swimming tecniques of frogs are as worthy as fishes. Their underwater motion is trust-drag based. By using the hydrodynamic equations of experimantal results of frogs' underwater swimming, we obtain the speed and the distance for such a motion.
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