Abstract: This paper presents a new algebraic method for investigation of extreme distan-ces of robot arm.The algebraic expressions for determination of the following twoclasses of extreme distances are derived:(1)extreme distance between two arbi-tary points located at hand and frame coordinate systems respectively;(2)extrememutual perpendicular distance line between two arbitrary lines located at hand andframe coordinate systems respectively.By using these algebraic expressions the pro-blem of the extreme distances of robot arms with an arbitrary number of joints canbe solved easily.