Application of mechanical advantage and instant centers on singularity analysis of single-dof planar mechanisms Jo urnal o f Me c hanic al Eng ine e ring , Vol. ME 41, No . 1, June 2010 Tra nsa c tio n o f the Me c h. Eng . Div., The Institutio n o f Eng ine e rs, Ba ng la de sh 50 A PPLIC A TIO N O F M EC HA NIC A L A DV A NTA G E A ND INSTA NT C ENTERS O N SING ULA RITY A NA LYSIS O F SING LE- DO F PLA NA R M EC HA NISM S Soheil Zarkandi* Department of Mechanical Engineering Babol University of Technology, Mazandaran, Iran *Corresponding email: slzarkandi@yahoo.com Abstract: Instantaneous kinematics of a mechanism becomes undetermined when it is in a singular configuration; this indeterminacy has undesirable effects on static and motional behavior of the mechanism. So these configurations must be found and avoided during the design, trajectory planning and control stages of the mechanism. This paper presents a new geometrical method to find singularities of single-dof planar mechanisms using the concepts of mechanical advantage and instant centers. Key Words: Planar mechanisms; Singularity; Instant center; Angular velocity; Mechanical advantage. INTRODUCTION The concept of instantaneous center of rotation (instant center) was introduced by Johann Bernoulli 1 . Instant centers are useful for velocity analysis of planar mechanisms and for determining motion transmission between links 2 . The method has proved to be efficient in finding the input-output velocity relationships of complex linkages 3 . Another application of instant centers is singularity analysis of planar mechanisms. Different approaches have been adopted in dealing with singularity analysis of planar mechanisms; considering a mechanism as an input–output device, Gosselin and Angeles 4 identified three types of singularities: Type (I) singularities (inverse kinematic singularities) occur when inverse instantaneous kinematic problem is unsolvable. This type of singularities occurs when at least one out of the input-variable rates can be different from zero even though all the output-variable rates are zero. In one- dof mechanisms, such singularities occur when the output link reaches a dead center, i.e. when an output variable reaches a border of its range; also for this type of mechanisms, in type (I) singularities mechanical advantage becomes infinite because in this configurations, at least one component of output torque (force), applied to the output link, is equilibrated by the mechanism structure without applying any input torque (force) in the actuated joints. Type (II) singularities (direct kinematic singularities) occur when direct instantaneous kinematic problem is unsolvable. This type of singularities occurs when at least one out of the output-variable rates can be different from zero even though all the input-variable rates are zero. In one- dof mechanisms, such configurations occur when the input link reaches a dead center. In type (II) singularities, a (finite or infinitesimal) output torque (force), applied to the output link, need at least one infinite input torque (force) in the actuated joints to be equilibrated, which in one-dof mechanisms, corresponds to a zero mechanical advantage. Type (III) singularities (combined singularities) occur when both the inverse and the direct instantaneous kinematic problems are unsolvable, i.e. when two previous singularities occurs simultaneously; In this type of singularities the input–output instantaneous relationship, used out of such singularities, holds no longer and the mechanism behavior may change. In one-dof mechanisms, these singularities lead to one or more additional uncontrollable dofs. Many articles have been presented for singularity analysis of single 5-6 or multi 7-10 dof planar mechanisms; some of these articles have geometrically addressed the singularity analysis of planar mechanisms using instant centers. For instance, Daniali 10 classified singularities of 3-dof planar parallel manipulators through instant centers. Di Gregorio 6 presented an exhaustive analytical and geometrical study about the singularity conditions occurring in single-dof planar mechanisms, which is based on the instant centers. He also 9 found singular configurations of multi dof planar mechanisms, considering the n dof mechanisms as the union of n one dof planar mechanisms and using the principle of superposition. There is a close relation between singularities of a one-dof mechanism and its stationary configurations; in this case, Yan and Wu 11, 12 gave a geometric criterion to identify which instant centers coincide at a stationary configuration 11 and developed a geometric methodology to generate planar one-dof mechanisms in dead center positions 12 . Here the author presents a method for singularity analysis of one-dof planar mechanisms using the concepts of mechanical advantage and instant centers.