ASME DETC An Introduction to Screw System Theory Kenneth J. Waldron Ohio State University Vijay Kumar University of Pennsylvania

ASME DETC Introduction Geometric representation of spatial velocity fields and force systems Allows use of geometric reasoning l Facilitates visualization l Often extremely efficient l Informs algebraic development Instantaneous representation only l Position information must come from elsewhere l In some circumstances screw systems are position invariant

ASME DETC Basic Ideas The screw system of a kinematic connection represents all motions permitted by that connection in a given position The wrench system of the connection represents all wrenches that can be passively transmitted by the connection The screw system and wrench system of the connection are uniquely related by reciprocity

ASME DETC Order The order of a screw system is identical to the number of degrees of freedom (connectivity) of the connection it represents l A screw system of order zero contains no screw axes l A screw system of order one is a single, unique screw axis l A screw system of order six includes all screw axes The order, c’, of the system reciprocal to a given screw system of order c is given by c’ = 6 - c A screw system of order c is uniquely defined by any c independent member axes

ASME DETC Serial and Parallel Connections The screw system of two connections acting in series is the combination of the systems of the individual connections The screw system of two connections acting in parallel is the intersection of the systems of the individual connections The reciprocal system of two connections acting in series is the intersection of the reciprocal systems of the individual connections The reciprocal system of two connections acting in parallel is the combination of the reciprocal systems of the individual connections

ASME DETC References l Waldron, K. J., Constraint Analysis of Mechanisms. J. Mechanisms Vol. 1, 1966: pages l Waldron, K. J., and Hunt, K.H. Series-Parallel Dualities in Actively Coordinated Mechanisms. Int. J. Robotics Research. Vol. 10. No. 5. Oct. 1991: pages l Kumar, V., Excerpts from MEAM 520 Lecture Notes, l Waldron, K. J., Excerpts from ME 851 Lecture Notes, 1998.

ASME DETC Outline Introduction [KJW, 10 mins] Background [VK, 15 mins] l Geometric definition l screw system of joints l open chains, closed chains What are Screw Systems? [VK, 20 mins] l Algebraic definition, screw coordinates, linear algebra l Examples l List of all screw systems l Important special screw systems u U-joint system u Spherical system u Planar system

ASME DETC Outline (continued) Reciprocity [VK, 20 mins] l Definition l Motion of a joint versus wrench transmitted by a joint l Other examples l Reciprocal screw systems Classification of Screw Systems [KJW, 20 mins] l Notation l Exhaustive list l Special screw systems Coffee Break

ASME DETC Outline (continued) Mechanisms [KJW, 20 mins] l Relationship of contact geometry to motion system l Over constrained mechanism examples Serial chain robots [VK, 30 mins] l Jacobian matrix and screws l Singularities l Working volume and singularity Parallel mechanisms, grasping, walking [KJW, 30 mins] l Jacobian matrix and screws l Force generating systems l Singularities l Series parallel duality

ASME DETC Introduction Ken’s introduction will come here