TAILIEUCHUNG - Advanced Robotics - Control of Interactive Robotic Interfaces Volume 29 Part 11

Tham khảo tài liệu 'advanced robotics - control of interactive robotic interfaces volume 29 part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 188 5 Transparency in Port-Hamiltonian Based Telemanipulation T Xe T x X TfrK. cotangent space at the point xe G Xe. T Xe T Xe cotangent bundle of Xe. Hence there exists an intrinsic duality product on Xe denoted by being a bilinear map from TXe X T Xe to R defined as e ve G If Xe v G T e Xe xe G Xe Since we want to use part of the internal energy to change parameters there is no need to change the power port structure. The power port structure therefore is still represented by the same m-dimensional vector space V and its dual V and their intrinsically defined duality product. We can therefore consider the following vector bundle see Sec. over the extended space manifold Xe. Qe TXe T xe v v The fibers of the bundle Qe define the space of extended vectors extended covectors flows and efforts. Q xe 7xe X Ifxe X V X V A Dirac structure can thus be defined on the extended state space manifold as a vector sub-bundle D . c Qe using Def. after having introduced -pairing on Qe analogously to what done in Sec. . It can be easily shown that dimD xe dim Xe dim V dim X dimK dim V n k m Furthermore being De a Dirac structure we have that ti le u f cl ìn u f d 0 VT u f d G . 1 ơeiơe w e J Ị 2 l e xe Both in fixed and variable parameters port-Hamiltonian systems the energy is a function is the same. The difference is that in the first case physical parameters are constant while in the second case parameters can change. Thus the Hamiltonian function for the extended case has the same structure of the Hamiltonian function for the fixed parameters case. Definition . Consider the extended state space Xe the space of flow variables V and dually the space of effort variables V the extended Dirac structure De and the Hamiltonian function H. Then the extended port-Hamiltonian system corresponding to the 4-tuple A e V De H is defined by setting A Passivity Preserving Tuning of Port-Hamiltonian Systems 189 ve xe ve dH dxe that is by

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