1. ASME DETC 1998 1 Screw Systems Vijay Kumar

2. ASME DETC 1998 2 Screw Systems Motivation l There is a need for adding and subtracting twists (screws). l Possible to define useful “systems of screws” by taking all linear combinations of joint twists. l Need to define the set of all possible twists (all possible instantaneous motions) of a rigid body. l Need to characterize all possible wrenches that can be resisted by a passive connection between two rigid bodies

3. ASME DETC 1998 3 What is a screw system? Kinematics The set of all instantaneous screw axis about which a body can (instantaneously) twist relative to another rigid body defines a screw system. Statics The set of all wrenches that can be exerted by one body contacting another body defines a screw system. Algebra The vector space of all screws (twists/ wrenches) generated by taking all possible linear combinations of a finite number of screws (twists/wrenches).

4. ASME DETC 1998 4 Screw Systems in Kinematics Two revolute joints in series l Screw axis for Joint 1 Axis 1 Axis 2 a1a1 a2a2 22 11 y z O l Screw axis for Joint 2 u2u2 u1u1 Link 0 Link 2

5. ASME DETC 1998 5 Screw Systems in Kinematics (Cont’d) Instantaneous motions of Link 2 relative to Link 0 l Defining twists: t 1 and t 2 l The screw system consists of all twists (t) about screws that are linear combinations of the defining screws. Axis 1 Axis 2 a1a1 a2a2 22 11 y z O u2u2 u1u1 Link 0 Link 2

6. ASME DETC 1998 6 Screw Systems in Statics The set of all wrenches that can be exerted on Body 2 by Body 1 l Defining wrenches: w 1, w 2, w 3 and w 4 l The screw system consists of all wrenches (w) about screws that are linear combinations of the defining screws: u Note we do not claim to know the intensities f i. We are interested in the set of all possible resultants. Body 1 Body 2 f1f1 f2f2 f3f3 f4f4

7. ASME DETC 1998 7 Screw Systems: Examples 1. Two rigid bodies connected by two revolute joints whose axes intersect The screw system consists of all screws (ISAs) with zero pitch, with axes in the plane of the two joint axes and passing through their intersection. Remarks: l The set of all twists is a two-dimensional vector space. We say that the screw system is order 2. l If the two joint axes had not intersected, the screw system would be much more difficult to characterize. BODY 3

8. ASME DETC 1998 8 Screw Systems: Examples 2. Two rigid bodies connected by three prismatic joints The screw system consists of all screws (ISAs) with infinite pitch. In other words, all translatory motions are instantaneously (in this particular example, also finite translatory motions) possible. Remarks: l The set of all twists is a three-dimensional vector space. We say that the screw system is order 3. l If the three joint axes had not been of infinite pitch, the screw system would have been much more difficult to characterize. Body 2 Body 1

9. ASME DETC 1998 9 Screw Systems: Examples 3. Two rigid bodies connected by three revolute joints with intersecting axes The screw system consists of all screws (ISAs) with zero pitch passing through the point of concurrence of the three joint axes. In other words, all rotations about the point of concurrence are instantaneously (in this case, also finite motions) possible. l If the axes are not all coplanar, the set of all twists is a three-dimensional vector space. We say that the screw system is order 3. Axis 4 Axis 5 Axis 6 P Q BODY 1 BODY 2

10. ASME DETC 1998 10 Screw Systems: Examples 4. Two rigid bodies connected by a spherical joint The screw system consists of all screws (ISAs) with zero pitch passing through the center of the spherical joint. In other words, all rotations about the point of concurrence are instantaneously (in this case, also finite motions) possible. l The set of all twists is a three-dimensional vector space. We say that the screw system is order 3. BODY 1 BODY 2

11. ASME DETC 1998 11 Screw Systems: Examples 5. Two rigid bodies with frictionless point contacts Body 1 can exert a pure force on Body 2 along the contact normal. The contact can transmit a zero pitch wrench whose axis is the contact normal. l The screw system associated with the contact wrenches is of order 1 l The intensity of the wrench may be limited (e.g., non negative and less than allowable maximum) l Note there is a completely different screw system associated with the allowable twists Body 2 Body 1 f1f1

12. ASME DETC 1998 12 Screw Systems: Examples 6. Two rigid bodies with frictional point contacts Body 1 can exert a pure force on Body 2 along any line passing through the contact point. The wrench system consists of all zero pitch wrenches with axes through the contact point. l The screw system associated with the contact wrenches is of order 3. l Note there is another screw system associated with the allowable twists (body 1 relative to body 2). Body 2 Body 1 f1f1 f3f3 f2f2

13. ASME DETC 1998 13 Screw Systems: Examples 7. A rigid body suspended by n strings. Body 1 can exert a pure force on Body 2 along any string. The wrench system consists of linear combinations of the four zero pitch wrenches defined by the four strings. l The screw system associated with the four wrenches is of order 4 (provided the four lines are linearly independent). l Note there is another screw system associated with the allowable twists (body 1 relative to body 2). Body 1 Body 2 f1f1 f2f2 f3f3 f4f4

14. ASME DETC 1998 14 Screw Systems: Examples 8. A spherical four-bar linkage. Each of the four revolute joints represents a zero pitch screw. These four screws define a screw system that consists of all zero pitch screws passing through O, the point of concurrence. l The screw system associated with the four defining zero pitch screws is of order 3. The four lines are linearly dependent. l The universal joint is an important application.

15. ASME DETC 1998 15 Screw Systems: Examples The Universal Joint The U-Joint has two orthogonal axes of rotation. l The drive axis and the driven axis together with the two U-joint axes form a spherical four bar linkage. l The four axes define a third order screw system consisting of all zero pitch screws passing through the center of the U-joint.

16. ASME DETC 1998 16 Screw Systems: Examples 9. Planar linkages All planar linkages have four or more joints. Each joint is either a revolute joint with the axis perpendicular to the plane or a prismatic joint in the plane. l The defining screws are of zero pitch perpendicular to the plane or infinite pitch parallel to the plane. l The four or more axes define a third order screw system consisting of all zero pitch screws perpendicular to the plane and all infinite pitch screws parallel to the plane.

17. ASME DETC 1998 17 Screw Systems: Examples 10. Overconstrained mechanisms