Joint clearance in mechanisms and robots leads to uncertainty in function deviation. Unlike the impact of the link tolerance on the performance quality, the uncertain effect of the joint clearance to the performance can not be eliminated by calibration because of the random nature. In this paper, based on the probability theory, a general probability density function for the output of planar robots is established for any probability density function of joint clearance. The result is demonstrated by a uniform distribution in the joint clearance and a table of the resulting functions is presented. These distribution functions and the table provide a convenient way to obtain the probability value for a planar robot to position its end point within a desired deviation zone and to determine the joint clearance value based on the concerned shape of tolerance zone and the specified probability value of repeatability.

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