Robotic Manipulator Control : Fundamentals of Task Space Design

space 1 The space of twist Joint space The space of wrench (6) jT T T J ~ ( = J KI Ii) I ' * 1 v torque space 1 Figure 2.6: Transformation from Task Domain to Joint Domain where each column of J is the image in abstract space of the corresponding column of the Jacobian J . Thus. the base of the joint variable space is: and the base of the joint torque space is: Because J is a matrix defined in the abstract space where orthogonality holds. the ort hogonality is meaningful in the joint spaces as well. Although the orthogonality is meaningful in both the abstract space and the joint space. the decomposition of constrained tasks is conducted in the screw space because tasks are naturally described t here. .As mentioned above, the decomposi tion is based on the reciprocity between the twist space in ray coordinates and the wrench s Pace in axis coordinates. As shown in Fig.7. the twist and wrench spaces are decomposed according to equations (2.51 ) to (2.54). Correspondingly. the abstract space is decornposed into two subspaces (2.23) which are orthogonal to each other. It has been shown that the orthogonality between &-, and R, is inherited by the reciprocity between the freedom twist subspace and the constraint wrench subspace and between the constraint twist subspace and the freedom wrench subspace. In the joint variable and torque spaces. the orthogonality holds between the freedom joint variable subspace and the constraint joint torque subspace and between the constraint joint variable subspace of and the freedom joint torque subspace as shown in equations (2.99) to (2.102). It should be noted that orthogonality between t h e images of &-m and R, in the joint and torque spaces can not be guaranteed because