Tham khảo tài liệu 'control of redundant robot manipulators - . patel and f. shadpey part 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 190 Appendix B Trajectory Generation Special Consideration for Orientation Figure Block diagram of the open-loop simulation for orientation TG. f k t k t k 0 A derivation of the above function is given below. The calculation of the angle-axis formulation from the DCM representation is as follows k t kx ky kzf k t 9 t ỈTÍ Ò where 9 t k t and k t 7-7. bft kx2ve c e x kxkyU0 - kzs9 kxkzvQ kys 0 R kxky u0 kzs 0 ky2u9 c 9 kykz u0 -kxs 0 kxkz e-kys 0 kykzU9 kxs 9 kz2u9 c 9 ax nx sx ay ny sy az nz sz where u9 1 - c 0 . This yields tr R 2c 0 1 where tr R ax ny sz _ vect R _ k - - where s 0 vect R 2 nz - sy sx - az ay - nx_ Appendix B Trajectory Generation Special Consideration for Orientation 191 Now we differentiate with respect to time to get K t k t 9 t k t ê t We need to find k Q as a linear function of K . To do this we note that d vect R vecRR and 0 - m -1 z y R R Q z 0 - r _-roT x 0 So that vectRR vect QR 2x where X tr R I - R and tr R Tr QR -2s 9 kw Now yields _ _ i _ tr R I - R C t c 9k 0 k -------- ----------------- 2s 0 s9 Differentiating with respect to time results in é trR. -2 s9 Substituting into yields é k w From equations and we get 2 s 9 N-1 k where N QM 2s9kkT and M tr R I- R - 2c9kkT Substituting in from and results in 192 Appendix B Trajectory Generation Special Consideration for Orientation 2s ek M e 2 s 0 kkT F where F Me 2s 9 kkT Differentiating yields 2 c ek 2s ek F ữ F ữ F-1 2c9 k 2 s 9 k - F Now we need to find F T T -T. F Me Mẽ 2ceẽkk 2s9 kkT kk where M - 2s9k I - 2s9k ữkkT - 2c9 kkT kk The optimized C code for this function is produced by the symbolic optimization routine provided by the RDM software 78