A.2 Screw Theory Appendix: Geometric Terminology A.1 Linear Spaces and Convexity

An ensemble of screws is known as a screw system, and is deened by a set of n 6 independent basis screws. The order of a screw system is equal to the number of basis screws required to deene it; such a system is also called an n-system. The order of a screw system reciprocal to an n-system is (6 ? n). With an innnitesimal rigid motion of an object in three-dimensional Euclidean space there is an associated screw called twist such that the body rotates about and translates along its screw axis. The screw coordinates of a twist are given by t = (T 1 , T 2 , T 3 , T 4 , T 5 , T 6), where the rst three components T 1 , T 2 and T 3 correspond to the angular displacement (or angular velocity), !, of the body and the last three components T 4 , T 5 and T 6 correspond to the translational displacement (or translational velocity), v, of a point xed in the body and lying at the origin of the coordinate system. The pitch of the twist is given by p = ! v ! ! : The pitch of the twist is the ratio of the magnitude of the velocity of a point on the twist axis to the magnitude of the angular velocity about the twist axis. If the pitch of a twist is zero then the twist corresponds to a pure rotation, and if the pitch of a twist is innnite then the twist corresponds to a pure translation. The magnitude of the twist is given by jtj = 8 < : k!k 2 ; if p < 1; kvk 2 ; if p = 1. Similarly, with any system of forces and torques acting on a rigid object in three-dimensional Euclidean space there is an associated screw called wrench such that the system of forces and torques can be replaced by an equivalent system of single force along the wrench axis and a torque about the same wrench axis. The screw coordinates of a wrench are given by w = (W 1 , The pitch of the wrench is the ratio of magnitude of the torque acting about a point on the axis to the magnitude of the force acting along the axis. If the pitch of a wrench is zero then the wrench …