Tham khảo tài liệu 'fundamentals_of_robotic_mechanical_systems part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 116 4. Kinetostatics of Simple Robotic Manipulators respectively. It is hence convenient to represent the foregoing relations in each individual frame which can be readily done by means of similarity transformations. Indeed if we apply the transformations b to each of aj . and Qi . respectively we obtain aj or correspondingly Qj in ĩị. Therefore eq. becomes Q. . Q. . Q. . Q. . Q. . Q. . Q . Now for compactness let US represent Q . simply by Q and let US recall the abbreviated notation introduced in eq. whereby Qj j is denoted simply by Qj thereby obtaining Q. Q. Q. Q. Q. Q. Q Likewise eq. becomes a. Q. a. Q. a. Q. Q. a. Q. Q. Q. a. Q. Q. Q. Q. a. p in which both sides are given in base-frame coordinates. Equations b above can be cast in a more compact form if homogeneous transformations as defined in Section are now introduced. Thus if we let Tj Tj j be the 4x4 matrix transforming Tị. .-coordinates into T coordinates the foregoing equations can be written in 4 X 4 matrix form namely . T with T denoting the transformation of coordinates from the end-effector frame to the base frame. Thus T contains the pose of the end-effector. In order to ease the discussion ahead we introduce now a few definitions. A scalar vector or matrix expression is said to be multilinear in a set of vectors Vj if those vectors appear only linearly in the same expression. This does not prevent products of components of those vectors from occurring as long as each product contains only one component of the same vector. Alternatively we can say that the expression of interest is multilinear in the aforementioned set of vectors if and only if the partial derivative of that expression with respect to vector Vj is independent of Vj for i Ỉ . N. For example every matrix Qj and every vector aj defined in eqs. and respectively is linear in vector Xj where Xj is defined as . . Xi 4-n Moreover the product Q. Q. Q. Q. Q. Q. appearing in .
