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Tham khảo tài liệu 'parallel manipulators new developments part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Wire Robots Part II Dynamics Control Application 141 The theorem for integration on manifolds states f Xi o H y det DH T DH dX ỊìoH Jdet DH T DHy dX A 19 20 where H A Ả H Ả is a map from A to . and DH is the Jacobian of H which is equal to H itself since it is linear. Furthermore jdet HT H is independent from Ả and can hence be canceled in the next step. Additionally splitting A into the simplexes gives 21 Since H is independent from Ả it can be moved out of the integral 22 Using eqn. 19 and eqn. 17 this can be rewritten as 23 Therefore fs HẢ holds where Ải denotes the CoG of A in R r. 3.3.2 Continuity of solution In this section the continuity of of the solution of the developed algorithm in the p-norm . p 1 is proven i.e. the function r R m n R n which maps a matrix Ae R mxn considered as a vector in Rm n onto the center of gravity as described before is continuous on the set of points of the workspace. 142 Parallel Manipulators New Developments Proof Again without loss of generality w 0 is assumed. First r is splitted into two mappings Ker R mn R n r and GravC R n r R n. The latter maps a vector p from R n r onto the center of gravity of the manifold spanned by the r n-dimensional downwards listed vectors in p. Ker R mn R nr maps a matrix A on its kernel H represented as a vector p in R nr. In calculations the kernel is still denoted with H for simplicity. Continuity of Ker and GravC implies continuity of r since r GravC Ker. First the continuity of GravCwill be proven. Therefore A A 0 is assumed i.e. the intersection of hypercube c and subspace is non-empty and thus also the CoG exists since continuity inside oR is to be proven. The CoG Ả s is considered fXidX X. . V A 24 1 t r Let Xs be the CoG of A where A is the preimage of F which is obtained from H H E. The matrices H hl . . . hr T e R nxr and E ei . . . er T e R nxr vectors in R n r. Then the p-norm of H is II pn In . fh r T . It follows are considered as lim A .i x .i lim IIEIIp-ol I II Blip-.0 lim .
