tailieunhanh - Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic Part 6

Tham khảo tài liệu 'multi-arm cooperating robots- dynamics and control - zivanovic and vukobratovic part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Mathematical Models of Cooperative Systems 87 where see Appendix B M is the inertial matrix and Yi col ria Ala G R6x1 Y col Y0 . Ym G R6m 6 Lva Y diag Lv0a Lvma G R Lvia diag T3x3 Lmia Aia G R6x6 Via coi fia Mia Aia Lvia Yi Yi G R6x1 Mia Lmia Aia Âia G R3x1 Wa Y diag W0a Wma LTa Y MLva Y G R 6m 6 x 6m 6 105 The overall potential energy due to linear and rotational displacements of the body is defined by mm 1 mm 1 n. E E 2yTK-yij E E 2 Yi - n - j-yj 106 i 0 j i 1 i 0 j i 1 where see Appendix B det nij 0 and n Kji Alj Yl Yj KljaAlj Yl Yj 107 ili w cK. cZ cY. G R6x6 ft 1 ij0H diag cijp cijP cijP cij cij cij G R p ự r r lự In combined form we have 2Ha YTna Y Y det na 0 rankna 6m 108 where na Y is a symmetric matrix na Y nT Y defined by na Y m 2 n0k k 0 k 0 n01 n01 n02 n0m n0m m E n1k k 0 k 1 n12 n1m G R 6m 6 x 6m 6 n1m n2m nkm k 0 k m 109 88 Multi-Arm Cooperating Robots The overall dissipation energy exchanged in the course of linear and rotational displacements of the body is defined by m m 1 1 El Ạ 1 . T _ _ 1 T _ _ . . _ . 22 2 Y - YjfDjY - Yj -2YTDa Y Y 110 i 0 j i 1 where D. DT D li o G . r- m Ds- . mA. DA e R1 A Ds. DA Dij Dij Dji lag Gija ia lja Dij Gija ia jap Dij e R Dij Dij is the damping matrix of elastic interconnections between the i th and jth nodes and as Dij Dji Da Y DT Y 111 n 22 D0k D01 D02 - - . D0m k 0 k 0 n D01 Ẻ D1k D12 D IS 1m . . k 0 k 1 . . . . . . . . . . D0m D1m D2m . . n . t D - k 0 k m - Generalized forces for the individual components Yj of the vector Yi are Qja L Gk mkS Fck k 0 Yi Gj mig Fj i 0 1 . m j 1 . 6. 112 By introducing 104 108 110 and 112 into the Lagrange equations 93 and after uniting all 6m 6 equations the general form of the model of the elastic system that under the action of the system of external contact forces Fc performs a macro motion the mobile unloaded state 0 will be Wa Y Y wa Y Y F 113 where see Appendix B Wa Y WT Y diag W0a Y0 . Wma Ym e R6 6 . det Wa Y 0 Wa Y Y col W0a Y Y . Wma Y Y e R6m 6. .