1. ASME DETC 1998 1 An Introduction to Screw System Theory Kenneth J. Waldron Ohio State University Vijay Kumar University of Pennsylvania http://www.cis.upenn.edu/~kumar/SST.html

2. ASME DETC 1998 2 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

3. ASME DETC 1998 3 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

4. ASME DETC 1998 4 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

5. ASME DETC 1998 5 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

6. ASME DETC 1998 6 References l Waldron, K. J., Constraint Analysis of Mechanisms. J. Mechanisms Vol. 1, 1966: pages 101-114. 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 473-480. l Kumar, V., Excerpts from MEAM 520 Lecture Notes, 1997. l Waldron, K. J., Excerpts from ME 851 Lecture Notes, 1998.

7. ASME DETC 1998 7 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

8. ASME DETC 1998 8 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

9. ASME DETC 1998 9 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

10. ASME DETC 1998 10 Introduction Ken’s introduction will come here