From higher degrees of shakiness to mobility

In this paper it is shown that a double-planar platform mechanism is architecturally mobile if in the given position it is shaky (singular) to the fourth degree. If a rigid construction in a singularity position admits an arbitrary state of velocity (acceleration, jerk, hyperjerk), we call it shaky to the first (second, third, fourth) degree. At a certain degree of shakiness any construction will become mobile; the double-planar platform mechanism (the anchor points of the actuators in the basis and in the platform lie in planes) is mobile starting at the fourth degree of shakiness. In a singularity position a construction is said to be infinitesimally mobile, although, it is actually immobile if all backlashes (which in reality are not totally avoidable) are excluded. A backlash of a certain magnitude causes a finite mobility of equal order in a non-shaky construction. In this paper it is shown that the effect of only one small backlash in one of the actuators of the platform mechanism rapidly grows with the degree of shakiness. If the platform mechanism is shaky to the fourth degree, it is architecturally mobile, i.e. it is mobile if we start with fixed actuator-lengths from any architecturally possible position. Motor-derivations in the sense of von Mises‘ motor calculus are used to find the equations that determine the positions of the anchor point of the actuators in a straightforward manner.