tailieunhanh - Multi-Arm Cooperating Robots- Dynamics and Control - Zivanovic and Vukobratovic part 15

Tham khảo tài liệu 'multi-arm cooperating robots- dynamics and control - zivanovic and vukobratovic part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Appendix B 271 nij nji cf. c 1 -Ịlpj011. 2 0 0 0 0 0 ij l r a-rja II 0 ct 1 - Jp j011 II 2 0 0 0 0 ij lira-rja 11 0 0 cz 1 jl 2 0 0 0 1 y llria rja II 0 0 0 ct. 0 0 ij 0 0 0 0 cj 0 0 0 0 0 0 cf. ij Therefore after substitution the total potential energy is 2na yT n01 n02 n0m Y0 - YTn01Y1 - Y0n02Y2 - - Yon0mYm - Yln10Y0 Y1T n10 n12 n1m Y1 - Yln12Y2 - - Y1 n1mYm Yminm0Y0 - Yminm1Y1 - Ym nm0 nm1 nm m-1 Ym or in comprised form 2na Y Tna Y Y 374 where due to nij nj na Y nT Y I x n 2_k 0 k 0 n0k -n01 -n02 -n0m E n _ k 0 k 1 n1k -n12 -n1m . En -rr k 0 k m nkm-I From this the derivative of potential energy with respect to the coordinate is ana dY 1 dYTnaY 2 dY na Y Y ana dY - T 3Y0 . . . ana dYt ana L dYm r 1 dYTnaY 2 3Y0 1 dYTnaY 2 dYi 1 dYTnaY -2 dYm n0a Y Y nta Y Y nma Y Y ana 1Y 1 dYTnaY 2 dYi nia Y Y G R6 272 Multi-Arm Cooperating Robots where nia Y G R 6x 6m 6 are the submatrices composed of the rows starting from 6i 1 to 6i 6 inclusive of the matrix na Y and d YTHaY dYi is the vector of the quadratic form scalar derivative YTnaY with respect to the vector Yi whereby the macron designates that partial derivation is carried out over the matrix na. In expanded form this vector is - dYTnaY -9Yi1 - Yt Y 9Yi1 dYTnaY dYTkaY YTdn Y dYi 9Yi2 dYTnaY L 9Y 6 J dY2 yT y L 9Yi6 Total dissipation energy consumed in the course of linear and rotational displacement of the body i relative to the body j i j 0 1 . m is defined by Da D01a D02a D03a D0ma D12a D13a D1ma D23a D2ma D m-1 m. An arbitrary member Dija of that sum is given by D T TAổ D T A -2Dija oij DijOij Aia Aja Dij Aia Aja ria r ja Gija ria rja Dij Gija ria rja ria r ja A ia A ja T DA A ia A ja ria rja T Aia Aja T x Gija ria rja Dj Gija ria rja 03x3 1 r ria rja 03x3 DA _ Aia A ja _ Yi Yj TDij Yi Yj Y TDijYi 2Ỹ DijỸj YjDjYj where D DT D li B G r .r DS. L l .n. . DA G R6x6. ij ij ji uiag Gija r ia r ja ij Gija ria r ja ij G Appendix B 273 wherefrom after substitution the total dissipation energy is determined by -2Da Y 0 D01