Scaling Direct Drive Robots

Recent experimental and analytical evidence indicates that direct drive robots become very practical and economical at miniature and microscopic scales, so it is interesting to understand quantitatively the properties of direct drive robots under scaling transformations. This leads to a study of how screws and their dual co-screws behave under the group of similarity transforms. This group is the group of isometries together with dilations. Several different representations are found on the space of screws and complementary representations are found on the dual space of co-screws. From the electromagnetic theory of the force and torque on a magnet in a magnetic field, we derive the scaling properties of the electromagnetic wrench. Hence, these results can be directly applied to the scaling of direct drive motors [1]. We conclude by proposing a scale-invariant measure for direct drive actuator performance.