tailieunhanh - Parallel Manipulators New Developments Part 4

Tham khảo tài liệu 'parallel manipulators new developments part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Conserving Integrators for Parallel Manipulators 81 To summarize the motion of the planar free rigid body is governed by the DAEs 1 with n 6 and m 3. This rigid body formulation is the cornerstone of the present approach to the energy-momentum integration of arbitrary multibody systems. Additional details about the present rigid body formulation may be found in BS01b BL06 . 4. Kinematic pairs This section deals with basic kinematic pairs which are fundamental for building complex multibody systems. Here we will present the revolute and the prismatic pair which represent the basic pairs necessary to model common planar parallel manipulators. Within this chapter we will also introduce a specific coordinate augmentation technique for both pairs in order to incorporate joint variables into the present rigid body formulation. The planar revolute pair Each rigid body of the multibody system depicted in Fig. 2 is modelled as constrained mechanical system as described in Section 3. Accordingly body A is characterized by 6 redundant coordinates 16 along with internal constraints I t qA e R 3 of the form 9 associated constraint Jacobian G t qA e R 3x6 of the form 15 and mass matrix MA e R 6x6 of the form 12 . Figure 2 The planar revolute pair. 82 Parallel Manipulators New Developments The description of the whole multibody system relies on the assembly of the individual bodies. The assembly procedure consists of the following steps. i The contributions of each individual body are collected in appropriate system vectors matrices. For example in the case of the present 2-body system Fig. 2 we get the vector of redundant coordinates q 17 along with the mass matrix M M1 06x6 .06x6 Af2 18 which in view of 12 is diagonal and constant. Moreover the constraints of rigidity are collected in the vector int ỉnt 19 with corresponding constraint Jacobian Gint Gint 03x6 3x6 G2nt 20 ii The interconnection between the rigid bodies in a multibody system is accounted for by external .