An Introduction to Screw System Theory

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

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. Order • The order of a screw system is identical to the number of degrees of freedom (connectivity) of the connection it represents • A screw system of order zero contains no screw axes • A screw system of order one is a single, unique screw axis • 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. 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. References • Waldron, K. J., Constraint Analysis of Mechanisms. J. Mechanisms Vol. 1, 1966: pages 101-114. • 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. • Kumar, V., Excerpts from MEAM 520 Lecture Notes, 1997. • Waldron, K. J., Excerpts from ME 851 Lecture Notes, 1998.

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

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

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

10. Introduction • Ken’s introduction will come here