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To satisfy the various demands such as no-backlash characteristic and so on, we have newly developed three types of no-backlash reducers:the cycloid ball reducer, the precession ball reducer and the reciprocating motion input type ball reducer. Especially, these reducers are developed for the robot joints. Also, these reducers are a kind of the constant velocity cam mechanisms with rolling balls; their inputs use eccentric motion, precession motion and reciprocating motion. The motion principle and the profile calculation method of each reducer are proposed, using the vector analysis.